Idempotenten in Groepringen

FACULTEIT WETENSCHAPPEN EN BIO-INGENIEURSWETENSCHAPPEN DEPARTEMENT WISKUNDE

Proefschrift ingediend met het oog op het behalen van de graad van Master in de Wiskunde Inneke Van Gelder

Promotor: Prof. Dr. Eric Jespers

MEI 2010 FACULTY OF SCIENCE AND BIO-ENGINEERING SCIENCES DEPARTMENT OF MATHEMATICS

Idempotents in Group Rings

Graduation thesis submitted in partial fulﬁllment of the requirements for the degree of Master in Mathematics Inneke Van Gelder

Promotor: Prof. Dr. Eric Jespers

MAY 2010 Voorwoord

De lente van 2005, nu exact vijf jaar geleden, een studiekeuze maken bleek plots een niet al te makkelijke opgave. Hoewel een vijf jaar durende studie eerst vreselijk lang leek, weet ik nu dat ik destijds de juiste keuze heb gemaakt. De afgelopen vijf jaren zijn immers voorbij gevlogen en een diploma komt nu duidelijk in zicht. ’Afstuderen’ is vaak een synoniem voor ’thesis schrijven’: D´eThesis. Een zware taak die ik vol spanning, maar toch ook met de nodige vrees, tegemoet ging. Die vrees was echter nergens voor nodig. Ik houd er een zeer leuke en leerrijke ervaring aan over. Alleen daarom al kan ik voor mezelf stellen dat ik tevreden ben met het resultaat. Nooit had ik gedacht zoveel te kunnen bereiken op enkele maanden tijd. Graag wil ik in het bijzonder mijn promotor bedanken. Eric Jespers, bedankt voor de goede begeleiding, motivatie en de vrijheid die je me gaf. Tevens bedankt dat je me de kans gaf gedurende een volledige week een conferentie bij te wonen. I would like to thank Angel´ del R´ıo.Thank you for your patient explanations, for your interesting lectures and talks and for the everlasting enthousiasm. And also to Gabriela Olteanu, for our interesting talks involving GAP en for your support. Tenslotte bedank ik ook mijn medestudenten voor de fantastische tijd die we gedurende vijf interessante en onvergetelijke jaren samen hebben beleefd.

Inneke Van Gelder Diest - mei 2010

i Inhoudsopgave

Inleiding 1

Samenvatting5

1 Preliminaries 14 1.1 Semisimple rings...... 14 1.2 Group rings...... 16 1.3 Representations and characters...... 19 1.4 Crossed products...... 22 1.5 Quaternion algebras...... 24

2 Primitive central idempotents of rational group algebras 25 2.1 Some deﬁnitions and background...... 25 2.2 The primitive central idempotent associated to a monomial character..... 30 2.3 Abelian-by-supersolvable groups...... 36 2.4 Example: metacyclic groups...... 41 2.5 An algorithm to compute the Wedderburn decomposition...... 42 2.6 More examples...... 49 2.7 The primitive central idempotent associated to an irreducible character.... 61

3 Primitive central idempotents of ﬁnite group algebras 65 3.1 Some notations and background...... 65 3.2 The primitive central idempotents...... 66 3.3 Examples...... 72

4 A complete set of orthogonal primitive idempotents 74 4.1 The rational group ring of nilpotent groups...... 74 4.2 Examples...... 86 4.3 Finite group rings of nilpotent groups...... 93

5 Appendix: some GAP functions 98

References 106

Index 110

ii Inleiding

De groepentheorie kent een lange geschiedenis1 die teruggaat tot de 17de eeuw, met wortels in het oplossen van algebra¨ısche vergelijkingen en getaltheorie. Toen was er echter nog geen sprake van het concept groep. Het artikel, Réflections sur la résolutionalgébriquedes équations, uit 1770 van Joseph- Louis Lagrange, gevolgd door artikels van Paolo Ruffini en Niels Henrik Abel, trok de aandacht van vele wiskundigen naar het concept permutaties. In 1830, was Evariste´ Galois de eerste om groepen en deelgroepen van permutaties te gebruiken. Hij hanteerde daarvoor ook de term groep, zoals in de monderne betekenis, maar dan beperkt tot permutaties. Hij introduceerde ook concepten zoals normale deelgroep en oplosbare groep. Augustin Louis Cauchy begreep de relevantie van permutatiegroepen als onafhankelijk onderwerp en schreef er dan ook een reeks interessante artikels over in de periode 1844-1846. Be¨ınvloed door Cauchy, realiseerde Arthur Cayley zich dat de notie van een groep geformuleerd kan worden in een meer abstracte context en hij gaf de eerste definitie van een abstracte groep in 1854 [Cay54]. Dit artikel wordt door velen beschouwd als het beginpunt van de abstracte groepentheorie. Uiteindelijk werd de definitie van een abstracte groep zoals we die vandaag gebruiken, gegeven door Walther Franz Anton von Dyck, een student van Felix Christian Klein, in 1882 [vD82]. In 1897 schreef William Burnside het eerste boek volledig gewijd aan groepentheorie, namelijk Theory of Groups of Finite Order. Dit werk gold tientallen jaren als het standaardwerk over eindige groepen. De ringtheorie is iets recenter dan de groepentheorie. In 1837 publiceerde William Rowan Hamilton éénvan zijn belangrijkste artikels over de complexe getallen. De complexe getallen a + bi werden eerder al door Euler en Gauss bestudeerd, maar Hamilton was de eerste die ze expliciet definieerde als een geordend paar (a, b) uit het reëlevlak. Later zocht hij naar manieren om analoog een driedimensionale ruimte te creëren.Hij slaagde er echter niet in om een goede vermenigvuldiging te definiëren.Later werd zelfs getoond dat dit onmogelijk is. Uiteindelijk probeerde hij het met vier dimensies en creërdede quaternionen. Dat zijn elementen van de vorm a + bi + cj + dk, met a, b, c, d reëlegetallen en i, j, k formele basissymbolen. Ze worden componentsgewijs opgeteld en de vermenigvuldiging verloopt volgens de regel

i2 = j2 = k2 = ijk = −1.

E´envan de bekenste anekdotes in de geschiedenis van de wiskunde is die van Hamilton die op 16 oktober 1843 samen met zijn vrouw over de Brougham Bridge in Dublin wandelt. Terwijl

1Deze korte geschiedenis komt uit [MS02].

1 Inleiding zijn vrouw tegen hem praat, luistert hij eigenlijk niet, maar denkt na over de vermenigvuldiging van viertallen reëlegetallen en ontdekt de quaternionen, de eerste niet-commutatieve ring ooit bestudeerd. Uit vrees om de ontdekking te vergeten, kerft hij de vermenigvuldigingsregel in een steen van de brug. Op die plaats is nu nog steeds een gedenkplaat met de welbekende formule te vinden. Datzelfde jaar introduceerde John Thomas Graves, ge¨ınspireerddoor zijn vriend Hamilton, de octonionen, een niet-associatieve uitbreiding van de quaternionen. Ze worden gedefinieerd als elementen van de vorm a0 + a1e1 + a2e2 + ··· + a7e7, met reëlecoëfficiënten ai en basiselementen ei. De optelling verloopt weer componentsgewijs en de vermenigvuldiging is gedefinieerd met enkele regels op de basiselementen. Het was echter Arthur Cayley die er in 1845 een eerste artikel over publiceerde, nadat hij de octonionen onafhankelijk van Graves ontdekte. Daarom zijn de octonionen ook gekend als de Cayley getallen. Hamilton breidde de constructie echter nog verder uit en definieerde de biquaternionen. Dat zijn opnieuw elementen van de vorm a + bi + cj + dk, maar met complexe coëfficiënten a, b, c, d. Kort daarna introduceerde hij de hypercomplexe systemen. Dat zijn dan weer verzamelingen van elementen van de vorm a1e1 + a2e2 + ··· + anen, waar de optelling componentsgewijs verloopt en de vermenigvuldiging gedefinieerd is op de basiselementen. Deze gebeurtenissen kunnen als de eerste stappen in de ontwikkeling van de ringtheorie gezien worden. Vanaf nu zou er meer interesse komen in de structuur van algebra’s. In 1870 gaf Benjamin Pierce een classificatie van alle gekende algebra’s van dimensie hoogstens 6. Hiertoe introduceerde Pierce enkele zeer belangrijke ideeënin ringtheorie, zoals de noties van idempotenten en het gebruik van idempotenten om een decompositie van een algebra te bekomen. In die eeuw vonden er ook belangrijke ontwikkelingen plaats in de theorie van niet- associatieve algebra’s. Volgend op het werk van Sophus Lie en Wilhelm Karl Joseph Killing in de studie van Lie-groepen en Lie-algebra’s, introduceerden Georg Scheffers en Eduard Study eind 19de eeuw enkele basisnoties voor de ontwikkeling in de theorie over de structuur van algebra’s, zoals de concepten van simpele en semisimpele algebra’s, hoewel ze een andere terminologie gebruikten. Deze resultaten inspireerden zowel Theodor Molien als Elie´ Cartan. Zij verkregen, onafhankelijk van elkaar, belangrijke resultaten over de structuur van eindigdimensionale reële en complexe algebra’s. Hiervoor introduceerden ze in deze context de noties van simpele en semisimpele algebra’s en karakteriseerden ze simpele algebra’s als volledige matrixalgebra’s. Dit werk bereikte zijn hoogtepunt in de mooie stellingen van Joseph Henry Maclagan Wedderburn, die de structuur van eindigdimensionale algebra’s over willekeurige lichamen beschrijven. Hiertoe gebruikte hij technieken gerelateerd met het bestaan van idempotente elementen, zoals gesuggereerd in het eerder werk van Pierce. Zoals eerder vermeld, gaf Cayley de eerste definitie van een abstracte groep. Het is inte- ressant om te vermelden dat in datzelfde artikel ook de notie van groepring verscheen voor de eerste keer. Hoewel hij precies dezelfde definitie gebruikte zoals we die de dag van vandaag hanteren, kende zijn werk weinig invloed en groepringen bleven nog een tijdje onbekend. Groepringen werden opnieuw ingevoerd door Molien wanneer hij zich realiseerde dat dit een natuurlijke setting was waarin hij sommige van zijn voorwaarden voor het semisimpel zijn, kon toepassen. Bovendien ontdekte Molien ondertussen enkele van de basisresultaten van de complexe representatietheorie van eindige groepen, inclusief de orthogonaliteitsrelaties van karakters.

2 Inleiding

De link tussen representatietheorie en de theorie over de structuur van groepalgebra’s, werd pas erkend na een invloedbare paper van Emmy Noether [Noe29], een artikel van Richard Brauer in 1929 [Bra29] en een artikel van beiden in 1927 [BN27]. Later kreeg het onderwerp op zichzelf meer belang nadat enkele vragen over groepringen werden ingesloten in Irving Kaplansky’s lijsten van open problemen in 1957 en 1970 [Kap57, Kap70]. Sindsdien verschenen boeken met een hoofdstuk over groepringen en in 1971 schreef Donald S. Passman het eerste boek volledig gewijd aan groepringen [Pas71]. Tot op heden verschijnen regelmatig artikels en boeken over groepalgebra’s. Er zijn immers nog veel problemen waarrond gewerkt wordt, waarvan we er enkele formuleren. Een eerste probleem is het isomorfismeprobleem, is het mogelijk dat twee niet-isomorfe groepen dezelfde integrale groepring hebben? Of anders geformuleerd, volgt uit ZG ' ZH dat G ' H? Dit probleem werd het eerst geformuleerd door Higman [Hig40]. Voor sommige klassen van groepen, bijvoorbeeld de eindig nilpotente groepen (Roggenkamp-Scott [RS87]), is het antwoord op het isomorfismeprobleem positief. In 2001 gaf Hertweck echter een tegenvoorbeeld voor het isomorfismeprobleem [Her01]. Maar er blijft nog steeds de vraag voor welke groepen het probleem wel een positief antwoord kent. In dit onderzoek speelt de eenhedengroep U(ZG) een grote rol, wat ons onmiddellijk bij het volgende probleem brengt. Men zoekt nog steeds naar generatoren voor een deelgroep van eindige index in U(ZG)[RS89, RS91, JL93], constructie van vrije deelgroepen [HP80, MS98] en naar structuurstellingen voor U(ZG) voor sommige klassen van groepen G [JPdR+07]. Het is essentieel om voor dit onderzoek ZG te beschouwen als een Z-order in de groepalgebra QG. Tenslotte zijn de eindige groepalgebra’s van groot belang in de codetheorie. Zo is een cyclische code namelijk een ideaal in de groepalgebra van een cyclische groep [HB98]. Bovendien is men ook ge¨ınte- resseerd in groepcodes, dat zijn eenzijdige idealen in willekeurige eindige groepalgebra’s FG [Ber67, Mac69, Cha82, DL89, Sab93, SL95, RTR97, HB98, KS01]. Voor al deze problemen is het zeer belangrijk om de structuur van de groepring RG goed te kennen. Onder bepaalde voorwaarden op de groep G en ring R, kan de groepring RG geschreven worden als een direct product van matrixringen. Dit is de welbekende Wedder- burn decompositie. Voor een eindige groep G en de rationale getallen Q, voldoet de groepring QG aan die voorwaarden en kan daarom geschreven worden als een direct product van matrixringen. Voor een eindig lichaam F en een groep G zodat char(F) - |G|, voldoet ook FG aan die voorwaarden. Het is echter van groot belang om de concrete realisatie van zo’n decompositie te geven. In het bijzonder moet men eerst de primitieve centrale idempotenten bepalen en vervolgens de primitieve idempotenten en een volledige verzameling van elementaire matrices. Het doel van deze masterproef is om beide groepen van idempotenten te onderzoeken, vandaar de titel Idempotenten in Groepringen. We behandelen alle bekende resultaten op dit gebied en bewijzen ze volledig. We geven ook een nieuw resultaat over het beschrijven van de orthogonale primitieve idempotenten in eindige semisimpele groepalgebra’s. In Hoofdstuk1 bundelen we de nodige voorkennis. Dit omvat theorie over semisimpele ringen, groepringen, groepsrepresentaties, kruisproducten en quaternionenalgebra’s, gebaseerd op [CR88, Isa76, MS02, Pas89, Pie82, Rei75]. In Hoofdstuk2 beschrijven we de primitieve centrale idempotenten, geassocieerd met monomiale irreduciebele karakters, van de rationale groepalgebra QG over een eindige groep G. We zien dat voor abels-bij-superoplosbare groepen alle primitieve centrale idempotenten een eenvoudige vorm aannemen, alsook de bijhorende Wedderburn componenten. Bovendien bekijken we ook de algoritmes om deze idempotenten en componenten te bepalen en gebruiken

3 Inleiding we de implementatie ervan in GAP om voorbeelden te geven. Op het einde van het hoofdstuk beschrijven we de primitieve centrale idempotenten van willekeurige rationale groepalgebra’s. Dit hoofdstuk is gebaseerd op [CKO+09, GAP08, JOdR10, OAdR03, OdRS04]. In Hoofdstuk3 zullen we de technieken van Hoofdstuk2 gebruiken om de primitieve centrale idempotenten en de Wedderburn componenten van semisimpele eindige groepalgebra’s FG over abels-bij-superoplosbare groepen G te beschrijven. Ook hier gebruiken we de implementatie in GAP om voorbeelden te geven. Voor dit hoofdstuk refereren we naar [BAdR07, CKO+09]. In Hoofdstuk4 beschrijven we, gebruikmakend van de resultaten in Hoofdstuk2, een volledige verzameling van orthogonale primitieve idempotenten van de rationale groepalgebra QG over een eindige nilpotente groep. Hiervoor maken we gebruik van [JOdR10]. Voor de voorbeelden gebruiken we een eigen implementatie in GAP. Vervolgens beschrijven we een volledige verzameling van orthogonale primitieve idempotenten van semisimpele eindige groepalgebra’s FG over nilpotente groepen G. Dit is een nieuw resultaat. In Hoofdstuk5 geven we tenslotte een overzicht van alle zelfgeschreven GAP-functies die gebruikt werden in de voorbeelden.

4 Samenvatting

In deze masterproef geven we een overzicht van alle recente resultaten over de primitieve centrale idempotenten en volledige verzamelingen van orthogonale primitieve idempotenten van semisimpele groepringen. Om te beginnen herhalen we in Hoofdstuk1 enkele belangrijke begrippen en eigenschappen. Eerst geven we in Sectie 1.1 enkele eigenschappen van semisimpele ringen, gebaseerd op [MS02]. Een links R-moduul M noemen we semisimpel als elk deelmoduul van M een directe sommand is. Een ring R noemen we dan semisimpel als die semisimpel is als links moduul over zichzelf. Een semisimpele ring is een directe som van een eindig aantal minimale linkse idealen (Stelling 1.3) en elk links ideaal is voortgebracht door een idempotent (Stelling 1.4). Daarom kunnen we die idempotenten gebruiken als karakterisatie van de decompositie in minimale linkse idealen, voor elke semisimpele ring bestaat er immers een volledige verzameling orthogonale primitieve idempotenten en omgekeerd bepaald zulk een volledige verzameling orthogonale primitieve idempotenten de decompositie van een semisimpele ring in minimale linkse idealen volledig (Stelling 1.5). We noemen een verzameling {e1, . . . , en} van idempotenten in een ring R een volledige verzameling van orthogonale primitieve idempotenten als de idempotenten optellen tot 1R, twee aan twee orthogonaal zijn en bovendien primitief zijn. Een idempotent wordt primitief genoemd als het niet de som is van twee niet-nulle orthogonale idempotenten. Het is dus reeds duidelijk dat de orthogonale primitieve idempotenten een belangrijke rol spelen in het bepalen van de structuur van een semisimpele (groep)ring. In de decompositie van een semisimpele ring als een directe som van minimale linkse idealen, kunnen we de isomorfe idealen samennemen. We verkrijgen op die manier een unieke decompositie van een semisimpele ring als een directe som van minimale tweezijdige idealen, die eveneens simpele ringen zijn (Stelling 1.7). We noemen deze componenten dan ook de simpele componenten. Bovendien weten we weer dat deze decompositie gekarakteriseerd wordt door idempotenten, er bestaat namelijk een unieke verzameling primitieve centrale idempotenten van de semisimpele ring (Stelling 1.9). Een verzameling primitieve centrale idempotenten is een verzameling {e1, . . . , en} van centrale idempotenten in een ring R, die optellen tot 1R, twee aan twee orthogonaal zijn en primitief zijn als centrale idempotenten. Hier is het duidelijk dat ook de primitieve centrale idempotenten zeer belangrijk zijn voor de structuur van semisimpele (groep)ringen. Dankzij Wedderburn-Artin weten we echter ook hoe elk zo’n simpele component eruit ziet. Elke simpele component is immers isomorf met een volledige matrixring over een scheef lichaam. Daarom verkrijgen we dat een semisimpele ring een direct product is van matrixringen over scheve lichamen (Stelling 1.10). Die decompositie is uniek en wordt de Wedderburn decompositie genoemd. In Sectie 1.2 herhalen we het concept van groepring, gebaseerd op [MS02]. Voor een ring RP en een groep G definiërenwe de groepring RG als de verzameling van de eindige sommen g∈G rgg, met rg ∈ R. De optelling is componentsgewijs en het product is gedefinieerd

5 Samenvatting

P P P P alsP g∈G rgg g∈G sgg = g,h∈G rgshgh. We definiërenook een product λ( g∈G rgg) = g∈G(λrg)g met λ ∈ R. Op deze manier is de groepring RG een vrij R-moduul met basis G. Als R bovendien commutatief is, dan is RG een R-algebra en spreken we ook van een groepalgebra. Als H een normale deelgroep is van G zodat |H| inverteerbaar is in R, dan kan het canonisch homomorfisme G → G/H uitgebreid worden naar eenP epimorfisme RG → f f −1 R(G/H) en bovendien is dan R(G/H) ' (RG)H, met H = |H| h∈H h. Het element Hf is trouwens een idempotent element in RG en als H normaal is dan is Hf zelfs centraal (Eigenschap 1.17). Dankzij Maschke weten we ook wanneer de groepring RG semisimpel is, namelijk wanneer R semisimpel is, G eindig en |G| inverteerbaar in R (Stelling 1.12). De rationale groepalgebra QG is dus semisimpel als G eindig is en de groepalgebra FG over een eindig lichaam is semisimpel als G eindig is en bovendien de karakteristiek van F en de orde van G relatief priem zijn. Voor deze groepringen kunnen we aldus de Wedderburn decompositie bepalen en zoals reeds vermeld leidt dit ons tot het bepalen van de primitieve centrale idempotenten. Voor Sectie 1.3 over representatietheorie baseren we ons op [Isa76] en [MS02]. Zij K een lichaam en A een K-algebra, dan is een representatie van A een algebramorfisme ρ : A → Mn(K), met n een positief natuurlijk getal dat we de graad van de representatie noemen. Twee representaties ρ, ψ van dezelfde graad n zijn equivalent als er een niet-singuliere n × n- matrix P bestaat zodat ρ(a) = P −1ψ(a)P voor alle a ∈ A. Met een representatie van A kunnen we een A-moduul associëren,en omgekeerd kunnen we met een A-moduul dat eindige rang over K heeft een representatie van A associëren. Twee modulen zijn isomorf als en slechts als hun geassocieerde representaties equivalent zijn. Irreduciebele representaties komen overeen met simpele modulen. Zij G een groep, dan is een K-representatie van G van graad n een homomorfisme ρ : G → GLn(K). Het is duidelijk dat K-representaties van G overeenkomen met representaties van KG door respectievelijk lineaire uitbreiding en beperking. Representaties bevatten echter nog te veel informatie omdat er verschil is tussen equivalente representaties. Daarom gaat men over naar karakters. Zij ρ een K-representatie van een groep G, dan is het K-karakter χ geassocieerd met ρ de functie van G naar K gegeven door χ(g) = trρ(g), waar tr staat voor het nemen van het spoor van de matrix. De graad van een karakter is χ(1) en karakters van graad 1 noemen we lineaire karakters. Zoals gewenst verkrijgen we nu dat karakters van equivalente representaties gelijk zijn (Lemma 1.23). Voor een lichaam F en een K-karakter χ van een groep G, noteren we F (χ) = F (χ(g) | g ∈ G), het karakterveld van χ over F . Een stelling van Brauer zegt dan dat, voor een complex karakter χ van een eindige groep G met exponent n, het karakterveld Q(χ) omvat is in Q(ξn) (Stelling 1.28). Als K algebra¨ısch gesloten is en G een eindige groep is zodat char(K) - |G|, dan zijn de primitieve centrale idempotenten van KG elk geassocieerd met een K-karakterP χ en we 1 −1 noteren dit e(χ). We berekenen deze idempotenten dan als volgt: e(χ) = |G| g∈G χ(1)χ(g )g (Stelling 1.27). Tenslotte definiërenwe nog ge¨ınduceerde karakters. Zij H een deelgroep van G G en zij χ eenP karakter van H, dan is het ge¨ınduceerde karakter χ van G gegeven door G 1 ◦ −1 ◦ ◦ χ (g) = |H| x∈G χ (xgx ), met χ (h) = χ(h) als h ∈ H en χ (y) = 0 als y∈ / H (Eigenschap 1.30). Voor Sectie 1.4 over kruisproducten gebruiken we [MS02], [Pas89] en [Rei75] als referen- tiewerken. Zij G een groep, R een ring en α : G → Aut(G) en τ : G × G → U(R) twee α afbeeldingen (respectievelijk de actie en twistingP genoemd), dan is het kruisproduct R ∗τ G gedefinieerd als verzameling eindige sommen i giri, met ri ∈ R, gi ∈ G en gi een symbool dat overeenkomt met gi. De optelling wordt componentsgewijs gedefinieerd en de vermenig-

6 Samenvatting

α vuldiging voldoet aan gh = ghτg,h en rg = gαg(r). Opdat R ∗τ G een associatieve ring zou zijn, moeten α en τ aan twee voorwaarden voldoen, die we hier niet zullen vernoemen. Merk op dat de groepring RG een bijzonder geval is van een kruisproduct. Als K een lichaam α is en α een trouwe actie, dan is K ∗τ G simpel (Lemma 1.32). Wanneer K/F een eindige Galoisuitbreiding is, G = Gal(K/F ) en α de natuurlijke actie is van G op K, dan noemen we α het kruisproduct K ∗τ G een klassiek kruisproduct en noteren we dit met (K/F, τ). Elk kruis- α product K ∗τ G met een trouwe actie kunnen we beschouwen als een klassiek kruisproduct. Bovendien is het klassiek kruisproduct (K/F, 1) isomorf met M[K:F ](F ) (Gevolg 1.34). Om Hoofdstuk1 af te sluiten, bundelen we nog enkele feiten over quaternionenalgebra’s in Sectie 1.5. Hiervoor gebruiken we het werk van [Pie82]. Zij F een lichaam met karakteristiek verschillend van 2 en a en b niet-nulle elementen in F . De vierdimensionale F -ruimte met basis 1, i, j, k en vermenigvuldiging gedefinieerd door de voorwaarden dat 1 een eenheidselement is 2 2 en i = a, j = b, ij = −ji = k, noemen we een quaternionenalgebra over F en noteren we a,b a,b F . Een quaternionenalgebra A = F is een scheef lichaam als en slechts als de vergelijking 2 2 2 x = ay + bz enkel kan gelden in F voor x = y = z = 0. Anders is A isomorf met M2(F ) en zeggen we dat A splitst (Eigenschap 1.37). In Hoofdstuk2 bestuderen we de primitieve centrale idempotenten van de rationale groepalgebra QG over een eindige groep G. In Sectie 2.1 geven we de nodige achtergrond. De klassieke methode voor het bepalen van de primitieve centrale idempotenten is eerst de primitieve centrale idempotenten e(χ), geassocieerd met de irreduciebele karakters χ van G, van CG bepalen en daarna de elementen van de vorm e(σ ◦ χ), met σ ∈ Gal(Q(χ)/Q), optellen (Sectie 2.1(2.1)). We noteren deze idempotenten eQ(χ) en bovendien is eQ(χ) het enige primitief centrale idempotent e van QG zodanig dat χ(e) 6= 0 (Lemma 2.1). In deze verhandeling zullen we ons echter concentreren op een nieuwere methode die noch van berekeningen met het karakterveld, noch met diens Galoisgroep, gebruik maakt. Deze methode werd ge¨ıntro- duceerd door Jespers, Leal en Paques [JLP03] voor nilpotente groepen en vervolgens uitgebreid en vereenvoudigd door Olivieri, del R´ıoen Simón[OdRS04] voor een veel ruimere klasse van groepen, inclusief de monomiale groepen. Deze methode beperkt zich tot een studie van de deelgroepen van G. Hierin spelen de centrale idempotenten (G, N) een cruciale rol. Voor een groep G en een normale deelgroep N, definiërenwe (G, N) als GÜ indien G = N en als Q f Ý M/N∈M(G/N)(N −M) indien G 6= N, waarbij M(G/N) de verzameling van minimaal normale niet-triviale deelgroepen van G/N is (Sectie 2.1(2.2)). Voor een eindige abelse groep G, zijn de primitieve centrale idempotenten van QG precies alle elementen van de vorm (G, N), met N een normale deelgroep van G zodat G/N cyclisch is (Eigenschap 2.4). De deelgroepparen die belangrijk zullen zijn, zijn de zogenaamde Shoda paren. Een paar (H,K) van deelgroepen van G noemen we een Shoda paar als K normaal is in H, H/K cyclisch is en als g ∈ G en [H, g] ∩ H ⊆ K, dan ook g ∈ H. De laatste voorwaarde garandeert dat als (H,K) een Shoda paar is van G en χ een lineair karakter is van H met kern K, dat dan χG een irreduciebel karakter is van G (Eigenschap 2.8). Zij H en K deelgroepen van G zodat K normaal is in H, dan definiërenweP e(G, H, K) als de som van alle G-geconjugeerden van (H,K), meer precies t e(G, H, K) = t∈T (H,K) , met T een rechts transversaal van CenG((H,K)) in G. Het is duidelijk dat dit element centraal is in QG en indien de G-geconjugeerden van (H,K) orthogonaal zijn, dan is dit element ook een idempotent. Deze e(G, H, K)’s spelen een cruciale rol in het bepalen van de primitieve centrale idempotenten van de rationale groepring over (niet-abelse) groepen. In Sectie 2.2 tonen we dat de primitieve centrale idempotenten van QG, geassocieerd

7 Samenvatting met een monomiaal irreduciebel karakter, berekend kunnen worden gebruikmakend van de elementen e(G, H, K). Een karakter χ heet monomiaal als er een deelgroep H van G en een lineair karakter ψ van H bestaan zodanig dat χ = ψG. Een groep G noemen we monomiaal als alle irreduciebele karakters van G monomiaal zijn. E´envan de hoofdstellingen luidt als volgt: zij G een eindige groep, H een deelgroep van G, χ een lineair karakter van H en χG het ge¨ınduceerdekarakter van χ op G. Als χG irreduciebel is, dan is het primitieve centrale idempotent van G geassocieerd met χG gelijk aan e (χG) = [CenG((H,ker χ)):H] e(G, H, ker χ) Q Q [Q(χ):Q(χG)] (Stelling 2.11). Deze stelling heeft twee onmiddellijke gevolgen. Als (H,K) een Shoda paar is van G, dan bestaat er een unieke α ∈ Q, zodat αe(G, H, K) een primitief centraal idempotent is van QG (Gevolg 2.12). Een eindige groep is monomiaal als en slechts als elk primitief centraal idempotent van de vorm αe(G, H, K) is voor een Shoda paar (H,K) en een α ∈ Q (Gevolg 2.13). We zoeken nu naar voldoende voorwaarden opdat α = 1. Het zal blijken dat het voldoende is om Shoda paren te versterken tot sterke Shoda paren. Een sterk Shoda paar van G is een paar (H,K) van deelgroepen van G zodat K een deelgroep is van H en H normaal is in NG(K), de normalisator van K in G, H/K cyclisch is en bovendien maximaal g abels in NG(K)/K en zodat voor elke g ∈ G \ NG(K), (H,K)(H,K) = 0. Een sterk Shoda paar (H,K) van G is een Shoda paar en bovendien is e(G, H, K) een primitief centraal idempotent van QG (Eigenschap 2.16). In dit geval kunnen we ook de simpele component geassocieerd met dit primitief centraal idempotent bepalen. Zij (H,K) een sterk Shoda paar, σ dan is QGe(G, H, K) isomorf met M[G:NG(K)](Q(ξ[H:K]) ∗τ NG(K)/H), met een zekere trouwe actie σ en twisting τ. We kunnen dit ook anders schrijven als M[G:NG(K)](Q(ξ[H:K])/F, τ), met σ F het centrum van Q(ξ[H:K]) ∗τ NG(K)/H, bepaald door de Galoisactie σ (Eigenschap 2.17). Dit geldt echter enkel voor de simpele componenten realiseerbaar door een sterk Shoda paar. In het algemeen kunnen we dus niet de Wedderburn decompositie van elke groepring over een willekeurige (monomiale) groep beschrijven. In Sectie 2.3 tonen we aan dat we voor abels-bij-superoplosbare groepen G wel de Wedder- burn decompositie van de groepring QG kunnen beschrijven, meer bepaald dat elk primitief centraal idempotent van QG van de vorm e(G, H, K) is met (H,K) een sterk Shoda paar van G (Stelling 2.25). Een groep G noemen we abels-bij-superoplosbaar als er een abelse normale deelgroep A bestaat zodat G/A superoplosbaar is. Als gevolg krijgen we dat een abels-bij- superoplosbare groep sterk monomiaal is (Gevolg 2.29). Een karakter χ van een groep G is sterk monomiaal als er een sterk Shoda paar (H,K) van G en een lineair karakter ψ van H met kern K bestaan zodanig dat χ = ψG. Een groep G noemen we sterk monomiaal als elk irreduciebel karakter sterk monomiaal is. In Sectie 2.4 bepalen we de primitieve centrale idempotenten van QG over een metacyclische groep G. Een metacyclische groep is een groep G met een normale cyclische deelgroep N zodat G/N cyclisch is. In Sectie 2.5 bespreken we algoritmes om (sterke) Shoda paren en hun geassocieerde primitieve centrale idempotenten te berekenen. Ook kunnen we de Wedderburn decompositie voor sterk monomiale groepen computationeel bepalen. Deze sectie is gebaseerd op [OAdR03]. De algoritmes zijn eveneens ge¨ımplementeerd in GAP in het pakket “Wedderga” door Cristo, Konovalov, Olivieri, Olteanu and del R´ıo[CKO+09, GAP08]. Het algoritme StrongShodaPairsAndIdempotents(QG) berekent de primitieve centrale idempotenten van QG, realiseerbaar door sterke Shoda paren. Indien de groep G sterk monomiaal is, verkrijgen we zo alle primitieve centrale idempotenten. Het algoritme ShodaPairsAndIdempotents(QG) bepaalt dan weer de primitieve centrale idempotenten van QG, realiseerbaar door Shoda

8 Samenvatting paren. Voor een monomiale groep, krijgen we aldus alle primitieve centrale idempotenten. We leggen het eerste algoritme in detail uit, ShodaPairsAndIdempotents verloopt analoog. Het idee is om op een slimme manier alle deelgroepen K van G te doorlopen en te testen voor welke deelgroepen H het paar (H,K) een sterk Shoda paar is. Met slim bedoelen we dan dat er geen overbodige groepen worden getest en dat we zo snel mogelijk alle goede paren vinden. Zo bekijken we telkens maar ´e´enrepresentant K van de conjugatiekassen van deelgroepen omdat als (H,K) een sterk Shoda paar is, dan ook (Hg,Kg) een sterk Shoda paar is en e(G, Hg,Kg) = e(G, H, K). Het algoritme stopt ook wanneer we een volledige lijst van primitieve centrale idempotenten gevonden hebben, dit wil zeggen wanneer de som van de idempotenten 1 is. Door Eigenschap 2.33, weten we ook dat er enkele onvermijdelijke sterke Shoda paren zijn van de vorm (G, K), met G0 ≤ K. Door de deelgroepen K van groot naar klein te doorlopen zullen we deze onvermijdelijke paren zo snel mogelijk kunnen ontdekken. Door Eigenschap 0 2.35 weten we ook dat h(NG(K)/K) ,Z(NG(K)/K)i cyclisch moet zijn, wat ons helpt om deelgroepen K uit te sluiten. Eigenschap 2.36 helpt ons om sneller kunnen over te gaan naar een andere conjugatieklasse. Ze zegt immers dat als (H,K) een paar is zodat K ≤ H¡NG(K), H/K cyclisch en maximaal abels in NG(K)/K is, dat dan e(G, H, K) volledig bepaald wordt door K. Op die manier lopen we door de conjugatieklassen van deelgroepen van G en vinden we uiteindelijk alle primitieve centrale idempotenten realiseerbaar door sterke Shoda paren. Het algoritme SimpleAlgebraByStrongSPInfo(QG, H, K) berekent een verzameling van data die die de simpele algebra QGe(G, H, K) beschrijft voor een sterk Shoda paar (H,K) van G. De combinatie van StrongShodaPairsAndIdempotents en SimpleAlgebraByStrongSPInfo zal dan voor sterk monomiale groepen, bv. abels-bij-superoplosbare groepen, de volledige Wedderburn decompositie bepalen. Het is natuurlijk op het eerste zicht niet duidelijk dat een Wedderburn component bepaald wordt door een verzameling getallen. Maar omdat we zulk een component kunnen beschrijven als een matrixring over een klassiek kruisproduct, waar de Galoisuitbreiding zelfs abels is, kan dit wel. Zij immers (H,K) een sterk Shoda paar van G, dan bestaan er getallen (oi, βi, γi)1≤i≤m en (δij)1≤i

9 Samenvatting e(G, H, K) inderdaad de bouwblokken voor de constructie van de primitieve centrale idempotenten e van QG zijn, meer bepaald dat elke e een rationale lineaire combinatie is van e(G, H, K)’s, waar (H,K) loopt doorheen de sterke Shoda paren van deelgroepen van G [JOdR10]. Dit bestuderen we in Sectie 2.7. Cruciaal is hier Brauers stelling over ge¨ınduceerde karakters.P Elk complex karakter χ van een eindige groep G is een Z-lineaire combinatie G G χ = i aiθi , met ai ∈ Z en karakters θi ge¨ınduceerdvan lineaire karakters θi van elementaire deelgroepen Mi van G (Stelling 2.48). Met een elementaire deelgroep bedoelen we een deelgroep die het direct product is van een cyclische groep met een p-groep voor een priemge- tal p. Zo verkrijgen we dan dat voor een eindige groep G van orde n en een irreduciebel karakter χ van G, het primitieve centrale idempotent e (χ) van G, geassocieerd met χ, van P Q Q χ(1) [Q(ξn):Q(ψi)] de vorm e (χ) = ai e(G, Hi,Ki) is, met ai ∈ ,(Hi,Ki) sterke Q [Q(ξn):Q(χ)] i [G:CenG((Hi,Ki))] Z Shoda paren van deelgroepen van G en ψi lineaire karakters van Hi met kern Ki (Stelling 2.50). Op het eerste zicht hebben we echter geen controle op de coëfficiënten zodat deze beschrijving wellicht niet implementeerbaar is. In Hoofdstuk3 gebruiken we de technieken van Hoofdstuk2 om de primitieve centrale idempotenten en de Wedderburn decompositie van eindige semisimpele groepalgebra’s FG te bepalen voor o.a. abels-bij-superoplosbare groepen. Dit hoofdstuk is gebaseerd op [BAdR07]. In Sectie 3.1 leggen we eerst enkele begrippen en notaties vast. Het lichaam F is altijd van orde q = pm en G van eindige orde n. Maschke zegt dat FG semisimpel is als p en n relatief priem zijn. We veronderstellen dat alle karakters van G genomen worden in F, de algebra¨ısche sluiting van F. Zoals in Hoofdstuk2 definiëren we, voor een irreduciebel karakter χ van G, eF(χ) als het unieke primitieve centrale idempotent ePvan FG zodanig dat χ(e) 6= 0. Volgens [Yam73] hebben we dan weer de formule eF(χ) = σ∈Gal(F(χ)/F) σe(χ). De eenhedengroup ∗ m ∗ Zn van Zn ageert op G door m · g = g . Zij Q de deelgroep van Zn gegenereert door de klasse van q en beschouw dan de actie van Q op G gedefinieerd door restrictie van de vorige actie. De q-cyclotomische klassen van G zijn de orbieten van G onder de actie van Q op G. Als g ∈ G, dan noteren we de q-cyclotomische klasse die g bevat met Cq(g) en q q2 qo−1 Cq(g) = {g, g , g , . . . , g }, met o de multiplicatieve orde van q modulo de orde van g. Als G cyclisch is, dan is G isomorf met G∗, de groep van irreduciebele karakters van G. Bijgevolg ∗ is ook G cyclisch. We noteren dan C(G) = Cq(G) voor de verzameling van q-cyclotomische klassen van G∗ die generatoren van G∗ bevatten. We definiërennu analoog aan (G, N) en e(G, H, K) uit Hoofdstuk2 twee elementen die een belangrijke rol zullen spelen. Zij N een normale deelgroep van GPzodat G/N cyclisch is en C ∈ C(G/N). Neem dan een χ ∈ C en stel (G, N) = |G|−1 tr(χ(gN))g−1. In deze definitie is tr = tr het spoor C g∈G P F(ξ[G:N])/F van de Galoisuitbreiding (ξ )/ , meer bepaald, tr (α) = σ(α) F [G:N] F F(ξ[G:N])/F σ∈Gal(F(ξ[G:N])/F) voor α ∈ F(ξ[G:N]). De definitie van C (G, N) is bovendien onafhankelijk van de gekozen χ ∈ C. Voor een K normaal in H en H een deelgroep van G zodanig dat H/K cyclisch is en C ∈ C(H/K) stellen we eC (G, H, K) de som van de verschillende G-geconjugeerden van C (H,K). In Sectie 3.2 zien we dat voor zekere groepen G en deelgroepen H en K, de elementen C (H,K) en eC (G, H, K) opduiken als de primitieve centrale idempotenten. Zo zijn voor een eindige abelse groep G van orde n en een eindig veld F van orde q zodat (q, n) = 1, de primitieve centrale idempotenten van FG precies de elementen van de vorm C (G, N) voor een normale deelgroep N van G zodat G/N cyclisch is en een C ∈ C(G/N). Bovendien is FGC (G, N) isomorf met F(ξ[G:N]) (Eigenschap 3.2). Als gevolg geldt voor elke eindige groep G met normale deelgroep N zodat G/N cyclisch is en C ∈ C(G/N), dat C (G, N) een

10 Samenvatting primitief centraal idempotent is van FG en dat FGC (G, N) ' F(ξ[G:N]). Bovendien geldt er voor een andere D ∈ C(G/N) dat C (G, N) = D(G, N) als en slechts als C = D (Gevolg 3.3). Noteer nu de lokalisatie van Z aan p met Z(p). Dan kunnen we Fp identificeren met het residulichaam Z(p)/Z(p)p van Z(p). We noteren het beeld van x ∈ Z(p) in Fp met x en breiden deze notatie uit tot de projectie van Z(p)G op FpG. We definieerden (G, N) voor N ¡ G als een element in QG, maar in principe behoort dit element tot Z(p)G, alsook e(G, H, K) voor K ¡ H ≤ G. Daarom is (G, N) een idempotent in FpG en als (H,K) een sterk Shoda paar is in GP, dan is e(G, H, K) een centraal idempotent in FpG. We verkrijgenP aldus dat (G, N) = C∈C(G/N) C (G, N) als G/N cyclisch is en dat e(G, H, K) = C∈R eC (G, H, K), voor K ≤ H ¡ NG(K) zodanig dat H/K cyclisch is en R een verzameling representanten is van de actie van NG(K) op C(H/K) door conjugatie (Lemma 3.5). Daaruit volgt dan bijna onmiddellijk dat voor een semisimpele groepring FG, een sterk Shoda paar (H,K) van G en C ∈ C(H/K), het element eC (G, H, K) een primitief centraal idempotent is van FG en dat FGeC (G, H, K) ' M[G:H](Fqo/[E:H] ), met E de stabilisator van eender welk element van C(H/K) onder de actie van NG(K) op C(H/K) en o de multiplicatieve orde van q modulo [H : K]. Bovendien geldt voor elke verzameling X van sterke Shoda paren dat elk primitief centraal idempotent van FG van de vorm eC (G, H, K) is met (H,K) ∈ X en C ∈ C(H/K), als elk primitief centraal idempotent van QG van de vorm e(G, H, K) is met (H,K) ∈ X (Stelling 3.6). Voor eindige abels-bij-superoplosbare groepen G zodat FG semisimpel is, verkijgen we dus dat elk primitief centraal idempotent van FG van de vorm eC (G, H, K) is voor een sterk Shoda paar (H,K) en C ∈ C(H/K) en dat FGeC (G, H, K) ' M[G:H](Fqo/[E:K] ), met E de stabilisator van eender welk element van C(H/K) onder de actie van NG(K) op C(H/K) en o de multiplicatieve orde van q modulo [H : K] (Gevolg 3.7). Omdat de voorgestelde techniek wederom algoritmisch is, werd deze eveneens ge¨ımple- menteerd in het GAP-pakket “Wedderga” [CKO+09, GAP08]. In Sectie 3.3 gebruiken we dit GAP-pakket om enkele voorbeelden te geven. Hoofdstuk4 is gewijd aan complete verzamelingen van orthogonale primitieve idempotenten. Zoals vermeld in Hoofdstuk2 zijn voor eindige abels-bij-superoplosbare (en dus in het bijzonder nilpotente) groepen G de primitieve centrale idempotenten van QG van de vorm e(G, H, K), met (H,K) een sterk Shoda paar van G en kunnen we de simpele component QGe(G, H, K) beschrijven. In Sectie 4.1 behandelen we voor nilpotente groepen G een betere en meer controleerbare beschrijving. Jespers, Olteanu en del R´ıobeschreven namelijk een volledige verzameling van matrixeenheden (in het bijzonder, een volledige verzameling van orthogonale primitieve idempotenten) van QGe(G, H, K)[JOdR10]. We baseren ons dan ook op dat artikel. Het is essentieel om hiervoor een volledige verzameling van orthogonale primitieve idempotenten van het klassiek kruisproduct (K/F, 1) te bepalen. Voor een lichaam K van karakteristiek 0, een Galoisuitbreiding K/F met G = Gal(K/FP ) en 1 |G| = n, en niet-nulle elementen x1, . . . , xn van K, vormen de geconjugeerden van |G| g∈G g met x1, . . . , xn een volledige verzameling van orthogonale primitieve idempotenten als en −1 slechts als trK/F (xixj ) = 0 voor alle i 6= j (Lemma 4.1). Niet elk kruisproduct met triviale twisting heeft echter een volledige verzameling√ van orthogonale primitieve idempotenten van deze vorm. Zij immers K = Q(ξ7) en F = Q( −7) en beschouw (K/F, 1), dan bestaan er geen x1, x2, x2 die aan de voorwaarden van Lemma 4.1 voldoen (Voorbeeld 4.3). Vervolgens is het ook essentieel om een classificatie te geven van eindige p-groepen G met een maximale abelse deelgroep die bovendien cyclisch is en normaal in G. We tonen dat G dan slechts 3 mogelijke presentaties kan hebben (Lemma 4.5). We willen ook een volledige verzameling van

11 Samenvatting orthogonale primitieve idempotenten van de quaternionenalgebra H(F ) kennen, voor F een lichaam van karakteristiek 0. De quaternionenalgebra splitst als en slechts als x2 + y2 = −1 1 1 voor x, y ∈ F . In dat geval vormen 2 (1+xi+yj) en 2 (1−xi−yj) een volledige verzameling van −1 orthogonale primitieve idempotenten van H(F ). Bovendien, als F = Q(ξm, ξ2n + ξ2n ) met m oneven, dan splitst H(F ) als en slechts als m 6= 1 en ofwel n ≥ 3 ofwel de multiplicatieve orde van 2 modulo m even is (Lemma 4.6). Met deze eigenschappen kunnen we een complete verzameling van orthogonale primitieve idempotenten van QG berekenen wanneer G eenQ nilpotente groep is. Een eindig nilpotente groep G kan steeds geschreven worden als G = p Sp, het direct product van de Sylow p-deelgroepen Sp met p | |G|. Het even deel vanQ een nilpotente groep G is de Sylow 2-deelgroep S2. Het oneven deel is het direct product p6=2 Sp, van alle Sylow p-deelgroepen met p 6= 2. Zij G een nilpotente groep, (H,K) een sterk Shoda paar van G, H/K = hai, N = NG(K) en stel N2/K en H2/K = ha2i (respectievelijk N20 /K en H20 /K = ha20 i) de even delen (respectievelijk oneven delen) van N/K en H/K respectievelijk. Dan heeft ha20 i een cyclisch complement hb20 i in N20 /K. Een volledige verzameling van orthogonale primitieve idempotenten van QGe(G, H, K) bestaat dan uit de geconjugeerden van Ý 2 [N20 :H20 ]−1 βe = b20 β2 door de elementen van Te = T20 T2TG/N , met T20 = {1, a20 , a20 , . . . , a20 }, TG/N een rechts transversaal van N in G en waar β2 en T2 worden ingevuld volgens vier gevallen (Stelling 4.7). De verschillende gevallen worden veroorzaakt door de drie mogelijke presentaties van eindige p-groepen G met een maximale abelse deelgroep die bovendien cyclisch is en normaal in G. Het vierde geval duikt op omdat twee scheve lichamen niet noodzakelijk terug een scheef lichaam voortbrengen, dit zou onder andere ook een matrixring over een scheef lichaam kunnen zijn. Omdat niet alle orthogonale primitieve idempotenten van een klassiek kruisproduct met triviale twisting zijn zoals in Lemma 4.1, kan deze methode niet zomaar uitgebreid worden naar bijvoorbeeld eindige metacyclische groepen. Bijvoorbeeld −1 2 voor G = C7 o C3 = hai o hbi, met b ab = a bestaat er geen volledige verzameling van orthogonale primitieve idempotenten van QGe(G, hai, 1) gevormd door Q(a)-geconjugeerden van eb(hai, 1). Het blijft dus een open probleem om een volledige verzameling van orthogonale primitieve idempotenten voor zulke groepen te construeren. Voor een nilpotente groep G kunnen we wel, voor een sterk Shoda paar (H,K) van G, een volledige verzameling van ma- 0 −1 0 trixeenheiden van QGeP(G, H, K) beschrijven als {Ett0 | t, t ∈ Te}, met Ett0 = t βet . Hiermee bedoelen we dat e = t∈Te Ett en Et1t2 Et3t4 = δt2t3 Et1t4 , voor elke t1, t2, t3, t4 ∈ Te. Bovendien is EttQGEtt ' F , in gevallen (1) en (2)(ii) van Stelling 4.7, en EttQGEtt ' H(F ), in geval (2)(i) van Stelling 4.7, met F het vaste deellichaam van QH(H,K) onder de natuurlijke actie van NG(K)/H (Stelling 4.11). In Sectie 4.2, geven we enkele voorbeelden van volledige verzamelingen van orthogonale primitieve idempotenten. Omdat Stelling 4.7 een algoritmische benadering is, kunnen we dit programmeren in GAP, meer bepaald in de functie OrthogonalPrimitiveIdempotents, die terug te vinden is in Hoofdstuk5. Er blijft echter wel een probleem aangezien we in geval (2)(ii) k k 2 2 [N20 :H20 ] 2 −2 van Stelling 4.7 wortels x en y van de vergelijking (1+x +y ) = 0 in Q(a20 , a2 +a2 ) 2n−2 2n−2 Ü 1+xa2 +ya2 c2 nodig hebben. In geval (2)(ii) is echter β2 = b2 2 met x en y wortels van de voorgaande vergelijking. Volgens Opmerking 4.8 kunnen we de verzameling ook bere- 2n−2 2n−2 1+xa2 +ya2 c2 kenen door bepaalde orthogonale idempotenten te vermenigvuldigen met 2 en 2n−2 2n−2 1−xa2 −ya2 c2 2 . Hoe dan ook, GAP kan x en y niet berekenen. Daarom zal onze GAP-functie een volledige verzameling van orthogonale primitieve idempotenten berekenen op de verme- 2n−2 2n−2 2n−2 2n−2 1+xa2 +ya2 c2 1−xa2 −ya2 c2 nigvuldiging met 2 en 2 na. Dit moet dan manueel gebeuren.

12 Samenvatting

Een voorbeeld wordt gegeven in Voorbeeld 4.16. Ook deze methode heeft dus beperkingen. Het is nu zeer voor de hand liggend om de voorgaande resultaten te hergebruiken om een volledige verzameling van orthogonale primitieve idempotenten voor een Wedderburn component in FG te construeren, voor F een eindig lichaam, G een nilpotente groep en zodanig dat char(F) - |G|. In Sectie 4.3 schrijven we onze bevindingen neer in Stelling 4.19. Dit is echter volledig analoog aan Stelling 4.7, alleen moeten we een geval minder beschouwen omdat we enkel te maken hebben met commutatieve lichamen. Eindige scheve lichamen zijn immers altijd commutatief en twee lichamen brengen opnieuw een lichaam voort. Toch was dit resultaat tot nu toe niet bekend en levert dus een originele bijdrage aan deze masterproef. Tenslotte geven we in Hoofdstuk5 ter vervollediging een overzicht van alle zelfgeschreven en gebruikte GAP-functies.

13 Chapter 1

Preliminaries

In this chapter, we collect some needed background. We do not always provide a proof, but we give references where additional information can be found. Unless explicitly stated otherwise, all rings are assumed to be associative and to contain an identity 1 and all modules are assumed to be left modules.

1.1 Semisimple rings

First of all, we recall some facts about semisimple rings and their Wedderburn decomposition, which can be found in [MS02].

Definition 1.1. An R-module M is called semisimple if every submodule of M is a direct summand.

Definition 1.2. A ring R is called (left-)semisimple if it is semisimple as a left module over itself. Similarly, one deﬁnes a right-semisimple ring. One can show that a ring is left- semisimple if and only if it is right-semisimple and therefore one simply speaks of semisimple rings.

Recall that the submodules of RR are precisely the left ideals of the ring R. Therefore R is semisimple if and only if every left ideal is a direct summand. One obtains the following theorem.

Theorem 1.3. Let R be a ring. Then, the following conditions are equivalent. (i) Every R-module is semisimple.

(ii) R is a semisimple ring.

(iii) R is a direct sum of a ﬁnite number of minimal left ideals.

One can say even more.

Theorem 1.4. Let R be a ring. Then R is semisimple if and only if every left ideal L of R is of the form L = Re, where e ∈ R is an idempotent.

14 Chapter 1. Preliminaries

Therefore, we can use the idempotents to characterize the decompositions of semisimple t rings as a direct sum of minimal left ideals. In other words, we get R = ⊕i=1Rei, with ei, 1 ≤ i ≤ t, idempotents of R. This is the classical Pierce decomposition.

t Theorem 1.5. Let R = ⊕i=1Li be a decomposition of a semisimple ring as a direct sum of minimal left ideals. Then, there exists a family {e1, . . . , et} of elements of R such that:

(i) ei 6= 0 is an idempotent element, 1 ≤ i ≤ t.

(ii) If i 6= j, then eiej = 0.

(iii) 1 = e1 + ··· + et.

0 00 0 00 0 00 (iv) ei cannot be written as ei = ei + ei , where ei, ei are idempotents such that ei, ei 6= 0 and 0 00 eiei = 0, 1 ≤ i ≤ t.

Conversely, if there exists a family of idempotents {e1, . . . , et} satisfying the conditions above, t then the left ideals Li = Rei are minimal and R = ⊕i=1Li. A set of idempotents satisfying the conditions of Theorem 1.5 is called a complete set of orthogonal primitive idempotents of the ring R. These idempotents will play the main role in Chapter4.

t Proposition 1.6. Let R = ⊕i=1Li be a decomposition of a semisimple ring R as a direct sum of minimal left ideals. Then every simple R-module is isomorphic to one of the ideals Li in the given decomposition. Given a decomposition of a semisimple ring R as a direct sum of minimal left ideals, re-ordering if necessary, we can group isomorphic left ideals together:

R = L ⊕ · · · ⊕ L ⊕ L ⊕ · · · ⊕ L ⊕ · · · ⊕ L ⊕ · · · ⊕ L . | 11 {z 1r1} | 21 {z 2r2} | s1 {z srs}

A1 A2 As

With Lij ' Lih and LijLkh = (0) if i 6= k. Also, it follows from Proposition 1.6 that all minimal left ideals are isomorphic to one of the ideals in the decomposition of R given above.

Theorem 1.7. With the notation above, let Ai denote the sum of all left ideals isomorphic to Li1, 1 ≤ i ≤ s. Then

(i) Each Ai is a minimal two-sided ideal of R.

(ii) AiAj = (0) if i 6= j.

s (iii) R = ⊕i=1Ai as rings, where s is the number of isomorphic classes of minimal left ideals of R.

Moreover, the ideals Ai, 1 ≤ i ≤ s, are simple rings. We also know that the decomposition of a semisimple ring R as a direct sum of minimal two-sided ideals is unique.

1.1. Semisimple rings 15 Chapter 1. Preliminaries

Definition 1.8. The unique minimal two-sided ideals of a semisimple ring R are called the simple components of R. Since a two-sided ideal is also a left ideal, we can conclude that for each simple component Ai, there exist an idempotent ei, such that Ai = Rei. We get a result similar to Theorem 1.5.

s Theorem 1.9. Let R = ⊕i=1Ai be a decomposition of a semisimple rings as a direct sum of minimal two-sided ideals. Then, there exists a family {e1, . . . , es} of elements of R such that:

(i) ei 6= 0 is a central idempotent, 1 ≤ i ≤ s.

(ii) If i 6= j then eiej = 0.

(iii) 1 = e1 + ··· + es.

0 00 0 00 (iv) ei cannot be written as ei = ei + ei , where ei, ei are central idempotents such that 0 00 0 00 ei, ei 6= 0 and eiei = 0, 1 ≤ i ≤ s. The idempotents in Theorem 1.9 will play the main role in Chapter2 of this paper and they are called the primitive central idempotents of R. So far, we know that a semisimple ring can be written as the direct sum of its simple components. Wedderburn and Artin give a precise description of these simple rings.

Theorem 1.10 (Wedderburn-Artin). A ring R is semisimple if and only if

R ' Mn1 (D1) ⊕ · · · ⊕ Mns (Ds), a direct product of matrix rings over division rings Di. Moreover, this decomposition is unique. The decomposition above is called the Wedderburn decomposition. Actually, one knows that each Mni (Di) ' Ai, following the notations above.

1.2 Group rings

In this section, we provide some information on group rings. For more details, we refer to [MS02]. Let G be a group and R a ring. We denote by RG the set of all formal linear combinations of the form X α = rgg, g∈G where rg ∈ R and rg = 0 almost everywhere, that is, only a finite number of coefficients are different from 0 in each of these sums. We define the sum of two elements in RG componentwise: X X X rgg + sgg = (rg + sg)g. g∈G g∈G g∈G

1.2. Group rings 16 Chapter 1. Preliminaries

The product of two elements in RG is defined by X X X X X rgg sgg = rgshgh = rxsy g. g∈G g∈G g,h∈G g∈G x,y∈G xy=g It is easily verified that RG becomes a ring with unit with the operations above. We can also define a product of elements in RG by elements λ ∈ R as X X λ rgg = (λrg)g. g∈G g∈G Again, it is easy to see that RG is an R-module. Actually, it is a free R-module with basis the group elements of G. Also, if R is commutative, then RG is an algebra over R.

Definition 1.11. The set RG, with the operations deﬁned above, is called the group ring of G over R. In the case where RPis commutative, RG is also called the group algebra of G over R. Given an element α = g∈G rgg of RG, the support of α is the subset of elements in G that appear eﬀectively in the expression of α, that is

supp(α) = {g ∈ G | rg 6= 0}. We introduce another notation. If R is a ring , H a ﬁnite subgroup of G and |H| invertible in R, we shall denote by Hf the following element of RG: 1 X Hf = h. |H| h∈H

The element X h, h∈H will be denoted by Hc. If H is a normal subgroup of G, then the canonical homomorphism ω : G → G/H can be extended to an epimorphism ωH : RG → R(G/H) such that X X ωH rgg = rgω(g). g∈G g∈G

The kernel of ωH is the left ideal of RG generated by the set {h − 1 | h ∈ H}, which we denote by ∆R(G, H). That is, ( ) X ∆R(G, H) = αh(h − 1) | αh ∈ RG . h∈H It can be shown that f f ∆R(G, H) = {α ∈ RG | αH = 0} = RG(1 − H), hence R(G/H) ' (RG)H.f (1.1) We will now determine necessary and suﬃcient conditions on R and G for the group ring RG to be semisimple.

1.2. Group rings 17 Chapter 1. Preliminaries

Theorem 1.12 (Maschke’s theorem). Let G be a group. Then, the group ring RG is semisimple if and only if the following conditions hold. (i) R is a semisimple ring. (ii) G is ﬁnite. (iii) |G| is invertible in R.

Corollary 1.13. Let G be a ﬁnite group and let K be a ﬁeld. Then, KG is semisimple if and only if char(K) - |G|.

Example 1.14. Let G be a ﬁnite group. The rational group algebra QG is semisimple. A translation of the Wedderburn-Artin Theorem in this context will give us plenty of information on the structure of a group algebra.

Theorem 1.15. Let G be a ﬁnite group and let K be a ﬁeld such that char(K) - |G|. Then each simple component Bi of KG is isomorphic to a full matrix ring of the form Mni (Di), where Di is a division ring containing an isomorphic copy of K in its center, and the isomorphism r KG ' ⊕i=1Mni (Di) is an isomorphism of K-algebras.

Corollary 1.16. Let G be a finite group and let K be an algebraically closed field such that char(K) - |G| (for example K = C), then r KG ' ⊕i=1Mni (K) 2 2 2 and n1 + n2 + ··· + nr = |G|, where r is the number of conjugacy classes of G. As we have seen, the problem of computing the Wedderburn decomposition of a semisimple group ring RG leads naturally to the problem of describing the primitive central idempotents of RG. In Chapter2, we will explain this in more detail for the rational group algebra QG. In Chapter3, we will do this for finite group algebras FG. Some central idempotents of RG are already known.

Proposition 1.17. Let R be a ring and let H be a ﬁnite subgroup of a group G. If |H| is invertible in R, then Hf is an idempotent of RG. Moreover, if H is normal in G, then Hf is central. Perlis and Walker gave a description for group rings of ﬁnite abelian groups.

Theorem 1.18 (Perlis-Walker). Let G be a ﬁnite abelian group, of order n, and let K be a ﬁeld such that char(K) - n. Then

KG ' ⊕d|nadK(ξd),

nd where ξd denotes a primitive root of unity of order d and ad = . In this formula, nd [K(ξd):K] denotes the number of elements of order d in G.

1.2. Group rings 18 Chapter 1. Preliminaries

1.3 Representations and characters

We give some basic information on representations and characters of a ﬁnite group G, based on [Isa76] and [MS02].

Definition 1.19. Let K be a field and A a K-algebra. A representation of A is an algebra homomorphism ρ : A → Mn(K). The integer n is the degree of ρ. Two representations ρ, ψ of degree n are similar if there exists a nonsingular n × n matrix P , such that ρ(a) = P −1ψ(a)P for all a ∈ A. It is easy to build modules from representations and representations from modules. If ρ is a representation of degree n of the K-algebra A, let V be the n-dimensional vector space over K. Define av = ρ(a)v for v ∈ V and a ∈ A. This gives the structure of an A-module to V . Conversely, if M is an A-module, which is of finite rank over K, choose a K-basis B for M and let ρ(a) be the matrix of the K-linear map M → M : m 7→ am with respect to the basis B. Moreover, Bρ(a) = aB. It is easy to check that ρ is a representation. Choosing another basis, will result in a similar representation. Starting with a representation ρ, constructing the module V as above, and then choosing the appropriate basis for V and constructing the corresponding representation will result in the original representation ρ. Let V and W be A-modules. Then, V and W are isomorphic if and only if the representations arising from V and W are similar. If V is an A-module and W < V is a proper nonzero submodule, choose a basis for W and extend this to a basis of V . Let ρ be the representation of A corresponding to V with respect to the constructed basis. Let ψ be the representation corresponding to W . It is then easy to see for a ∈ A that ρ(a) has the form ψ(a) θ(a) ρ(a) = , 0 ξ(a) where ξ is a representation corresponding to V/W and θ is a function from A into (n−m)×m matrices (where n = dim V and m = dim W ). However, θ is not a representation. The representation ρ is said to be in reduced form and one similar to ρ is reducible. Thus the irreducible representations correspond to the irreducible or simple modules. If there exists a submodule U < V in the above situation, with V = W ⊕ U, then the basis for W may be extended to V by adjoining to it a basis for U. When this is done, the result is that θ(a) = 0 for all a ∈ A. It follows that if ρ is any representation corresponding to a completely reducible module, then ρ is similar to a representation in block diagonal form, where each of the blocks is an irreducible representation. Let G be a finite group and let K be a field. Suppose ρ is a representation of KG with degree n. Since ρ is an algebra homomorphism, ρ(1) = I, the identity matrix. It follows for g ∈ G, that ρ(g)−1 = ρ(g−1). If we restrict ρ to G ⊂ KG, we obtain a group homomorphism from G into the general linear group GL(n, K).

Definition 1.20. Let K be a field and G a finite group. Then a matrix K-representation of G is a homomorphism ρ : G → GL(n, K) for some integer n.A K-representation of G is a homomorphism ρ : G → GL(V ), where V is a free K-module of finite rank. Using linear algebra, it is easy to see that there is a parallelism between K-representations and matrix K-representations.

1.3. Representations and characters 19 Chapter 1. Preliminaries

We have seen that a representation of KG determines a K-representation of G by restriction. Conversely, a K-representation ρ0 of G determines a representation ρ of KG by linear extension. That is, X X ρ( kgg) = kgρ0(g). g∈G g∈G We shall usually use the same symbol to denote both the K-representation of G and the corresponding representation of KG. Also, the adjectives “similar” and “irreducible” will be applied to K-representations of G as if they were the corresponding representations of KG. The trouble with representations is that they contain too much information. Some of the data is clearly redundant because it distinguishes between similar representations. The idea behind character theory is to throw away most of the information and to save just enough to be useful.

Definition 1.21. Let ρ be a K-representation of G. Then the K-character χ of G afforded by ρ is the function given by χ(g) = trρ(g), with tr the classic trace function. As the K-representations of KG are determined by the K-representations of G by linear extension, so are the K-characters of KG. For a K-character χ of G and a field F , the character field over F is defined as F (χ) = F (χ(g) | g ∈ G). We say that χ(1) = deg(ρ) is the degree of χ. Characters of degree 1 are called linear characters. These correspond with the group homomorphisms from G to K.

Definition 1.22. A mapping φ : G → K is called a class function if φ is constant on the conjugacy classes of G.

Lemma 1.23. Similar K-representations of G aﬀord equal characters and characters are class functions. If ρ and ψ are K-representations of G, then ρ(g) 0 ξ(g) = 0 ψ(g) is also a K-representation. Since trξ(g) = trρ(g) + trψ(g), it follows that the set of K- characters of G is closed under addition. We now restrict our attention to the special case where K is algebraically closed such that t char(K) - |G|, for example K = C. Fix a ﬁnite group G and write KG = ⊕i=1Li, a direct sum of simple KG-modules (i.e. minimal left ideals of KG). We also know that

KG = L ⊕ · · · ⊕ L ⊕ L ⊕ · · · ⊕ L ⊕ · · · ⊕ L ⊕ · · · ⊕ L . | 11 {z 1r1} | 21 {z 2r2} | s1 {z srs}

A1 A2 As

Choose a basis in each Li1 and let ρi be the resulting representation of KG. Let χi be the character aﬀorded by ρi. It follows that the set Irr(G) = {χ1, . . . , χs} is the set of all irreducible K-characters of G. In what follows, the word “character” means K-character, unless stated otherwise. The used algebraically closed ﬁeld K will beP clear from the context. s Since sums of characters are characters, it follows that χ = i=1 niχi is a character whenever the ni are nonnegative integers which are not all zero. Conversely, if χ is any

1.3. Representations and characters 20 Chapter 1. Preliminaries character of G aﬀorded by a representations ρ corresponding to a module V , we can decompose V into a direct sum of simple modules. It follows that χ is the sum of the corresponding irreducible characters. P s Let χ be a character of KG, then χ can be written as a sum i=1 niχi of irreducible characters χi of Li1, since similar representations aﬀord equal characters. s s We have not yet proved that the χi are distinct. We have KG = ⊕i=1Ai = ⊕i=1KGei, with 1 = e1 + ··· + es. Since Aj annihilates the module Li1 if i 6= j, we have ρi(ej) = 0 in this case. It follows that ρi(1) = ρi(ei) = I. Therefore, χi(ej) = 0 if i 6= j and χi(ei) = χi(1) 6= 0 and we conclude that the χi’s are distinct as functions on KG. Thus the χi’s are also distinct as functionsP on G. s If χ = i=1 niχi is a character, then those χi with ni > 0 are called the irreducible constituents of χ.

Definition 1.24. Let G be a ﬁnite group. The character ρ of G aﬀorded by a representation corresponding to the regular module KG, is called the regular character. This regular character ρ can be computed in two ways.

Lemma 1.25. Let G be a ﬁnite group and ρ the regular character of G. If g ∈ G and g 6= 1, then ρ(g) = 0. Also ρ(1) = |G|. Since ρ is a character of G, it may be expressed as an integer linear combination of the irreducible characters χi. We do this explicitly.

Lemma 1.26. Let GPbe a ﬁnite group and ρ the regular character of G. If Irr(G) = s {χ1, . . . , χs}, then ρ = i=1 χi(1)χi. Let χ be an irreducible character of the group G, then by the primitive central idempotent of KG associated to χ we mean the unique primitive central idempotent of KG such that χ(e) 6= 0, and we denote it by e(χ). Now we can compute the primitive central idempotents e(χ) of the group algebra KG explicitly for all irreducible characters χ of G.

Theorem 1.27. Let G be a finite group, K an algebraically closed field with char(K) - |G| and χ ∈ Irr(G). Then, e(χ), the primitive central idempotent of KG associated to χ, is given by 1 X e(χ) = χ(1)χ(g−1)g. |G| g∈G The next Theorem is a simplified version of Brauer’s Splitting Theorem [CR88, Th. 41.1].

Theorem 1.28. Let G be a ﬁnite group, let n be the exponent of G and let χ be a complex character of G. Then the character ﬁeld Q(χ) is contained in Q(ξn). Proof. Note that it is enough to show that for every g ∈ G, the corresponding character value χ(g) is a sum of roots of unity whose orders are divisors of n. First assume that G is abelian and χ is irreducible, then χ is linear. Since gn = 1, also χ(g)n = 1 and hence χ(g) is a root of unity whose order is a divisor of n.

1.3. Representations and characters 21 Chapter 1. Preliminaries

NowP assume that G is abelian and χ is an arbitrary character. Then χ can be written as k a sum i=1 miψi, where ψi are irreducible characters and mi ∈ N. Hence it follows that χ(g) is a sum of roots of unity whose order is a divisor of n. Now we consider the general case. Take a g in G. Consider the CG-module M associated to χ. Denote Mg the module M considered as a Chgi-module and denote ψ the character associated to Mg. Then it follows that ψ(g) is a sum of roots of unity whose order is a divisor of n. Since M and Mg have the same C-basis, the characters are unchanged. This proves the statement. 2

Let H be a subgroup of a group G. Given a character χ of G, we obtain the character χH of H by restriction. Now we induce characters of G from characters of H.

Definition 1.29. Let G be a group, H a subgroup of G and let φ be a class function of H. Then φG, the induced class function on G, is given by 1 X φG(g) = φ◦(xgx−1), |H| x∈G where φ◦ is deﬁned by φ◦(h) = φ(h) if h ∈ H and φ◦(y) = 0 if y∈ / H. Note that if T is a right transversal of H in G, then we can write X φG(g) = φ◦(tgt−1). t∈T

Proposition 1.30. Let G be a ﬁnite group and H a subgroup of G. Suppose φ is a character of H. Then φG is a character of G. We formulate one last Theorem on faithful irreducible characters. A representation ρ of G is said to be faithful if ker ρ = 1. A faithful character χ of G is then the character associated to a faithful representation of G.

Theorem 1.31. If G has a faithful irreducible character, then Z(G) is cyclic.

1.4 Crossed products

In this chapter, we recall the natural concept of a crossed product R∗G. Details can be found in [MS02] and [Pas89]. Let G be a group and R a ring. Suppose that we are given two maps

α : G → Aut(R) and τ : G × G → U(R) called action and twisting (or factor system), satisfying the relations

τgh,xαg(τg,h) = τg,hxτh,x and τg,hαg(αh(r)) = αgh(r)τg,h,

1.4. Crossed products 22 Chapter 1. Preliminaries

α for all g, h, x ∈ G and r ∈ R. Then by the crossed product R ∗τ G, of G over R with twisting τ and action α, we understand the set of ﬁnite sums ¦X © giri | ri ∈ R, gi ∈ G , where gi is a symbol corresponding to gi, called an invertible homogeneous element of degree gi. Equality and addition are deﬁned componentwise and for g, h ∈ G and r ∈ R we have

gh = ghτg,h and rg = gαg(r).

α If we extend this rule distributively, it is easy to see that R ∗τ G is an associative ring, which contains R and has as a homogeneous R-basis the set G, a copy of G. α As a special case, if τg,h = 1 for all g, h ∈ G, we get the skew group ring R ∗1 G. In addition, if we have αg = 1 for all g ∈ G, we get the group ring RG. Crossed products arise naturally. For example, if G has a normalS subgroup N then picking a ﬁxed set of cosetL representatives {x} of N in G, we can write G = x xN as a disjoint union and hence RG = x xRN. So G/N = {x | x ∈ G/N} is an RN-basis for RG. Since N is −1 normal in G, x RNx = RN and x induces a conjugation automorphism αx on RN. We have a map α : G → Aut(RN) and for x ∈ G/N and r ∈ RN, we have

−1 rx = x(x rx) = xαx(r).

Furthermore, for x, y ∈ G/N, we have xN · yN = xyN, so x · y = xyτx,y for some τx,y ∈ N ⊆ U(RN). Thus we have a map τ : G/N × G/N → U(RN). α It follows that RG = RN ∗τ G/N. The following Lemma can be found in [Rei75, Th. 29.6].

α Lemma 1.32. Let K be a ﬁeld and A = K ∗τ G be a crossed product with a faithful action α. Then A is simple. Proof. Let I be a nonzero two-sided ideal of A, and let

x = g1r1 + ··· + gnrn ∈ I,

with x 6= 0 and n minimal. If n > 1, choose s ∈ R, with αg1 (s) 6= αg2 (s). This is possible −1 because α is faithful. Then x−sxαg1 (s) is a shorter nonzero element of I. This is impossible and hence necessarily n = 1, and I contains a unit of A. Therefore I = A. 2

α Let K be a field and A = K ∗τ G. If the action of G on K is faithful, then one may identify G with a finite subgroup of the automorphism group of K. If we write F = Fix(G), the fixed subfield, then it follows that K is Galois over F and G ' Gal(K/F ). We define a special case of a crossed product.

α Definition 1.33. A classical crossed product is a crossed product K ∗τ G, where K/F is a ﬁnite Galois extension, G = Gal(K/F ) is the Galois group of K/F and α is the natural α action of G on K. The crossed product K ∗τ G is denoted by (K/F, τ). It is easy to check that the center of (K/F, τ) is F . Using Lemma 1.32, we obtain that the classical crossed product (K/F, τ) is a central simple F -algebra, that is a ﬁnite-dimensional associative algebra, which is simple, and for which the center is exactly F .

1.4. Crossed products 23 Chapter 1. Preliminaries

α As consequence one can consider every crossed product K ∗τ G with faithful action as a classical crossed product (K/F, τ), with F the ﬁxed subﬁeld of the action α of G on K. By the Wedderburn Theorem, we know that a classical crossed product is isomorphic to a matrix ring over a division algebra. When the twisting is trivial, we know more. The following Corollary can be found in [Rei75, Cor. 29.8].

Corollary 1.34. Let K/F be a ﬁnite Galois extension and n = [K : F ]. The classical crossed product (K/F, 1) is isomorphic to Mn(F ).

1.5 Quaternion algebras

We collect some facts on quaternion algebras from [Pie82]. Throughout this section, F is supposed to be a ﬁeld whose characteristic is not 2.

Definition 1.35. Let a and b be non-zero elements of F . Let A be the four dimensional F -space with basis 1, i, j, k and the bilinear multiplication deﬁned by the conditions that 1 is a unity element, and i2 = a, j2 = b, ij = −ji = k. We assume A to be associative. Then A = a,b is called a quaternion algebra over F . F Note that Hamilton’s real quaternions occur as the special case = −1,−1 . H R a,b Lemma 1.36. For any non-zero a and b from F , F is a simple algebra whose center is F . As a consequence of the Wedderburn decomposition, we can state the following proposition.

Proposition 1.37. A quaternion algebra over F is either a division algebra, or else it splits a,b (i.e. it is isomorphic to M2(F )). Moreover, the quaternion algebra F is a division ring if and only if the equation x2 = ay2 + bz2 can hold in F only if x = y = z = 0.

1.5. Quaternion algebras 24 Chapter 2

Primitive central idempotents of rational group algebras

In this chapter we obtain the primitive central idempotents of the rational group algebra QG over a ﬁnite group G associated to a monomial irreducible character, using a method which does not involve computations with the character ﬁeld nor its Galois group. For abelian-by- supersolvable groups this method takes an easy form. These results can be found in [OdRS04]. We also take a look at some algorithms, which implements the computation of primitive central idempotents and the Wedderburn decomposition of the rational group algebra for abelian-by-supersolvable groups. This can be found in [OAdR03]. These algorithms are also implemented in the GAP-package “Wedderga” [CKO+09]. Finally, we describe the primitive central idempotents for arbitrary groups. This result can be found in [JOdR10]. Unfortunately, this description seems not to be implementable.

2.1 Some deﬁnitions and background

We start with fixing some notations. In what follows, any group G will always be assumed to be finite. The group Aut(C) of automorphisms of the complex numbers acts on CG by acting on the coefficients, that is X X σ · agg = σ(ag)g, g∈G g∈G for σ ∈ Aut(C) and ag ∈ C. Furthermore, Aut(C) acts on the set of characters of G by composition, σ · χ = σ ◦ χ, for σ ∈ Aut(C) and χ a character of G. Since any automorphism of the complex numbers fixes the rationals, Theorem 1.27 gives us that e(σ · χ) = σ · e(χ) for any irreducible character χ of G. Since χ(1) ∈ N, the stabilizers of e(χ) and χ coincide with {σ ∈ Aut(C) | σ fixes Q(χ)}, which is the group Gal(C/Q(χ)) of automorphisms of C which fix the character field Q(χ) of χ. Since, by Brauer’s Theorem (1.28), for every irreducible character ψ of G, Q(ψ) ⊂ Q(ξn), where n is the exponent of G, we get that e(ψ) ∈ Q(ξn)G. Hence the actions of Aut(C) can be restricted to Aut(Q(ξn)). Since Q(ξn)/Q is an abelian Galois extension, by the Fundamental Theorem of Galois Theory, also Q(χ)/Q is an abelian Galois extension and Gal(Q(χ)/Q) ' Gal(Q(ξn)/Q)/Gal(Q(ξn)/Q(χ)). Therefore, both the orbits of χ and e(χ)

25 Chapter 2. Primitive central idempotents of rational group algebras can be computed applying the elements of Gal(Q(χ)/Q) to χ and e(χ) respectively, and these orbits have [Q(χ): Q] elements. We deﬁne, following [Yam73], X X eQ(χ) = σ · e(χ) = e(σ · χ). (2.1) σ∈Gal(Q(χ)/Q) σ∈Gal(Q(χ)/Q)

Lemma 2.1. Let G be a ﬁnite group and χ a character of G. Then eQ(χ) is the unique primitive central idempotent e of QG such that χ(e) 6= 0.

Proof. Clearly eQ(χ) is a central idempotent, since e(ψ)P is a primitive central idempotent in CG for each irreducible character ψ of G. Set eQ(χ) = g∈G agg. Since σ(eQ(χ)) = eQ(χ) for all σ ∈ Gal(Q(χ)/Q), we obtain that σ(ag) = ag for all σ and all g in G. Since Q(χ)/Q is a Galois extension, ag has to be in Q for all g in G and therefore eQ(χ) is an element in QG. Also, by the orthogonality relations, χ(eQ(χ)) = χ(e(χ)) = χ(1) 6= 0. We prove that eQ(χ) is primitive. The primitive central idempotents of CG are precisely the e(ψ), where ψ ranges over all the irreducible characters of G. Therefore each central idempotent e of QG is of the form e = e(ψ1) + ··· + e(ψn), where ψ1, . . . , ψn are distinct irreducible characters of G. Suppose eQ(χ) is not primitive in QG, then there exist nonempty subsets I,J ⊂ Gal(PQ(χ)/Q), with I ∪ J =P Gal(Q(χ)/Q) and I ∩ J = ∅, such that eQ(χ) = e1 + e2, where e1 = τ∈I e(τ · χ) and e2 = τ∈J e(τ · χ).P Suppose 1 ∈ I and take σ ∈ J. Since e1 ∈ QG and σ ∈ Gal(Q(χ)/Q), we have e1 = σ(e1) = τ∈ I e(στ · χ). Hence e(σ · χ) appears in the summand of e1 and σ ∈ I, a contradiction. It remains to prove the uniqueness. Let ePbe another primitive central idempotent of QG k with the property that χ(e) 6= 0. Then e = i=1 e(ψi), with ψi irreducible characters of G. Because of the orthogonality relations and χ(e) 6= 0, there exists j ∈ {1, . . . , k} such that

ψj = χ. Therefore eeQ 6= 0. But since diﬀerent primitive central idempotents are orthogonal, this contradicts our assumption. 2 P f 1 Recall that if H is a subgroup of G, H = |H| h∈H h ∈ QG is an idempotent of QG which g is central if and only if H is normal in G. If g ∈ G, we write ge = hgi. If G 6= {1}, we denote by M(G) the set of all minimal normal non trivial subgroups of G and deﬁne Y (G) = (1 − MÝ). M∈M(G) By convention (1) = 1. f If N is a normal subgroup of G, then we obtain an isomorphism QGN ' Q(G/N) (see (1.1)). Let (G, N) denote the preimage of (G/N) under this isomorphism. Clearly, ( Nf if N = G (G, N) = Q f Ý f Q Ý (2.2) M/N∈M(G/N)(N − M) = N M/N∈M(G/N)(1 − M) if N 6= G. Note that both (G) and (G, N) are central idempotents of QG. For an abelian group G, the primitive central idempotents of QG have been described as elements of the form (G, N) in [JLP03]. We take a look at this. Let e be a primitive central idempotent of QG, then Ge = {g ∈ G | eg = e} Ý is a normal subgroup of G. Clearly eGe = e and thus e is a primitive central idempotent of Ý QGGe ' Q(G/Ge). We need two lemmas.

2.1. Some definitions and background 26 Chapter 2. Primitive central idempotents of rational group algebras

Lemma 2.2. Let G be a finite group and let e be a primitive central idempotent of QG. Then Ge = {1} if and only if (G)e = e. In particular, G has a faithful irreducible representation if and only if (G) 6= 0. f Proof. Clearly, for a subgroup N of G, eN = e if and only if N ⊆ Ge. Recall that (G) is a central idempotent in QG. Assume that Ge = {1}, then (G)e 6= 0 is a central idempotent in QGe. Since e is a primitive central idempotent, it is the only central idempotent in QGe, so (G)e = e. Conversely, assume (G)e = e, then (G) 6= 0. Thus eMÝ 6= e, for all M ∈ M(G). Hence, for all M ∈ M(G), M 6⊂ Ge. Since Ge is a normal subgroup and G is finite, Ge has to be {1}, for otherwise Ge should be either minimal itself or either contain a minimal normal subgroup. This proves the first part of the lemma. We prove the second part. Let {e1, . . . , en} be the primitive central idempotents of CG and

CGei = Li1 ⊕ · · · ⊕ Liri , a direct sum of simple CG-modules. The irreducible representation ρ : GP→ GL(Li1): g 7→ (αei 7→ αeig) is faithful if and only if Gei = {1}. Furthermore, n (G) i=1 ei = (G) 6= 0 if and only if there exists a primitive central idempotent ei of CG such that (G)ei 6= 0. Because ei is the only central idempotent of CGei, the last statement is true if and only if (G)ei = ei. This holds if and only if Gei = {1}, equivalently with ρ being faithful. 2

Lemma 2.3. Let G be a ﬁnite group. If (G) 6= 0 then Z(G) is cyclic. The converse holds if G is a ﬁnite nilpotent group. Proof. If (G) 6= 0, then by Lemma 2.2, G has a faithful irreducible representation and by Theorem 1.31, the center Z(G) is cyclic. Conversely, assume G is nilpotent and Z(G) = hgi is cyclic, say of size m. Since (1) = 1, one may also assume G 6= {1}. Let A1,...,An be the minimal normal subgroups of G. Since G is a nilpotent group, any nontrivial normal subgroup intersects the center non trivially. Thus each Ai is actually a minimal normal subgroup of Z(G). So each Ai has to be cyclic of o(g)/pi prime orderQ and central, that is Ai = hg i, with pi a prime divisor of o(g). It follows that n f (G) = i=1(1 − Ai) 6= 0. Indeed, if (G) = 0, then

m 2m (p−1)m Y 1 g p g p g p (G) = (1 − − − − · · · − ) = 0. p|m p p p p

In that case, the coeﬃcient of the identity in G should be 0. However this is not the case since Y p − 1 6= 0 p|m p and products k m P k1m j j kim p g p1 ··· g j = g i=1 pi P never become the identity since m is not a divisor of j kim . 2 i=1 pi

Now we can give a description of the primitive central idempotents in the rational group algebra of a ﬁnite abelian group.

2.1. Some definitions and background 27 Chapter 2. Primitive central idempotents of rational group algebras

Proposition 2.4. Let G be a finite abelian group. The primitive central idempotents of QG are precisely all elements of the form (G, N), with N a subgroup of G so that G/N is cyclic. In particular, if e is a primitive central idempotent of QG, then supp(e) is a subgroup of G, f and e is a Z-linear combination of idempotents of the form H, where H is a subgroup of G. Proof. It is easily verified that if A = hai is a finite cyclic group, then (A) is a central φ idempotent of QA and QA(A) = Q(a(A)) ' Q(ξ|A|): a(A) 7→ ξ|A|. Therefore, we get that QA(A) is a field and therefore (A) is a primitive central idempotent. In particular all elements (G, N) with N a subgroup of G so that G/N is cyclic are primitive central idempotents of QG. Now if e is a primitive central idempotent of QG, then, by Lemma 2.2 and Lemma 2.3, Ý (G/Ge)e = e and Z(G/Ge) = G/Ge is cyclic, where e denotes the image of e = eGe in Ý QGGe ' Q(G/Ge). By the above, we have that (G/Ge) is a primitive central idempotent of Q(G/Ge), hence e is a central idempotent in Q(G/Ge)(G/Ge). Thus, e = (G/Ge). Hence it follows that e = (G, Ge). This proves the first part of the statement. Now let e be a primitive central idempotent of QG. Then e = (G, N) for a subgroup N of G such that G/N is cyclic. In that case e is a Z-linear combination of elements of the form fg Þ NM1 ··· Mn, with Mi/N ∈ M(G/N), so Mi ∩ Mj = N. Let now Ti be a transversal for N in ff f fg Þ å Mi. Since NN = N, it is easy to see that NM1 ··· Mn = hN,T1,...,Tni, where hN,T1,...,Tni is a subgroup of G. Now we consider supp(e). Suppose M(G/N) = {M1,...,Mm} and Ti is a transversal for N in Mi. We claim that supp(e) = hN,T1,...,Tmi, a subgroup of G. Clearly supp(e) is a subset of hN,T1,...,Tmi. Since the Ti are transversals for N of groups which coincide only in N, terms of the form nt1 ··· tm ∈ hN,T1,...,Tmi cannot disappear in 0 0 0 the summand of e. Namely, it can be shown by induction on m that if nt1 ··· tm = n t1 ··· tm, 0 0 0 then necessarily n = n , t1 = t1, ..., tm = tm. 2

If χ is a linear character of G and H = ker χ, then G/H is cyclic as a ﬁnite subgroup of C∗. By Proposition 2.4, ωH ((G, H)) = (G/H) = (G/H, 1) is a primitive central idempotent of Q(G/H) and therefore (G, H) is a primitive central idempotent of QG. In fact one has the following.

Lemma 2.5. If χ is a linear character of G and H = ker χ, then eQ(χ) = (G, H). Proof. If [G : H] = n and g ∈ G is such that the image of g in G/H is a generator of G/H, i.e. G/H = hgHi, then every element of G has a unique form as gih for i ∈ {0, . . . , n − 1} and h ∈ H. Moreover, χ(gih) = ξi, where χ(g) = ξ is a primitive n-th root of unity. The n/p minimal subgroups of G/H are of the form Lp/H = hg Hi, where p is a prime divisor of n. n/p So Lp = hH, g i. For every prime divisor p of n, one has pX−1 f Ý 1 in/p χ(H − Lp) = 1 − ξ = 1, p i=0 Q n/p f Ý since ξ is a primitive p-th root of unity. Then, χ((G, H)) = p|n χ(H − Lp) = 1 and the lemma follows since eQ(χ) was deﬁned as the unique primitive central idempotent e such that χ(e) 6= 0 and (G, H) is a primitive central idempotent. 2

We ﬁx some notation. For elements g, h ∈ G, the commutator of g and h is [g, h] := −1 −1 g h gh. For all subgroups H1,H2 of a group G,[H1,H2] denotes the subgroup, called

2.1. Some definitions and background 28 Chapter 2. Primitive central idempotents of rational group algebras

the commutator subgroup, generated by all the commutators [h1, h2] for hi ∈ Hi. For a subgroup H of G and an element g ∈ G, we denote Hg := g−1Hg. The following Theorem is an old one of Shoda [Sho33].

Theorem 2.6 (Shoda). Let χ be a linear character deﬁned on a subgroup H of G. Then the induced character χG is irreducible if and only if for every g ∈ G \ H there is an element h ∈ H ∩ Hg such that χ(ghg−1) 6= χ(h). Proof. A proof can be found in [CR88, Cor. 45.4]. This uses properties of the so called intertwinning number of two KG-modules. 2

Due to this theorem, a pair of subgroups satisfying certain rules, is called a Shoda pair.

Definition 2.7. A pair (H,K) of subgroups of G is called a Shoda pair if it satisﬁes the following conditions: (S1) K ¡ H, (S2) H/K is cyclic, (S3) if g ∈ G and [H, g] ∩ H ⊆ K, then g ∈ H. Set Aut(C). Note that giving an Aut(C)-orbit of a linear character of a subgroup H of G is equivalent to give a pair of subgroups (H,K) of G satisfying (S1) and (S2). For χ a linear character of H, one can take K = ker χ and then (H,K) satisﬁes (S1) and (S2). Note that every other character in the Aut(C)-orbit also has K as kernel. Conversely, for a pair (H,K) satisfying (S1) and (S2), one can take all linear characters of H with kernel K. Those characters form an Aut(C)-orbit. Then Shoda’s Theorem can be rephrased as follows.

Proposition 2.8. If χ is a linear character of a subgroup H of G with kernel K, then the induced character χG is irreducible if and only if (H,K) is a Shoda pair. Proof. Assume that χ is a linear character of a subgroup H of G with kernel K such that χG is irreducible. From the remarks above we already know that (H,K) is a pair satisfying (S1) and (S2). We prove (S3). Let g ∈ G such that [H, g] ∩ H ⊆ K and assume that g∈ / H. Shoda’s Theorem 2.6 gives us an element h ∈ H ∩ Hg such that χ(ghg−1) 6= χ(h). Therefore h−1ghg−1 ∈/ K, but this contradicts our assumption since h−1ghg−1 ∈ [H, g] ∩ H. Conversely, assume that χ is a linear character of a subgroup H of G with kernel K and (H,K) is a Shoda pair. Take g ∈ G \ H and assume that χ(ghg−1) = χ(h) for all h ∈ H ∩ Hg and hence h−1ghg−1 ∈ K. Therefore [H, g] ∩ H ⊆ K and by (S3), g ∈ H, a contradiction. So, there exists an element h ∈ H ∩ Hg such that χ(ghg−1) 6= χ(h) and by Shoda’s Theorem, χG is irreducible. 2

We give an example of a Shoda pair.

Example 2.9. Let G be an abelian group. Then for every subgroup H of G such that G/H is cyclic, (G, H) is a Shoda pair. For example, if G = Z2 × Z2 and H = Z2, then (G, H) is a Shoda pair.

2.1. Some definitions and background 29 Chapter 2. Primitive central idempotents of rational group algebras

Let G and H be groups. The normalizer of H in G is defined as {g ∈ G | gH = Hg} and denoted NG(H). The group G acts on the right on CG by conjugation, i.e. x · g = xg := g−1xg, for x ∈ CG and g ∈ G. Moreover, G acts on the right on the set of characters of subgroups of G by defining χ · g, for χ a character of a subgroup H of G and g ∈ G, as the character of Hg given by (χ · g)(hg) = χ(h). Now take χ an irreducible character of H, a subgroup of G and ρ the associated representation. g ThenP consider thePCH -module V associated to χ · g. Recall that the action on V is defined g g as h∈H αhh v = h∈H αh(ρ·g)(h )v. It is easy to seeP that V is alsoP a CH-module because of the definition of ρ · g, more precisely with the action h∈H αhhv = h∈H αhρ(h)v. Note that this CH-module is precisely the module associated to the character χ, so V has to be simple as a CHg-module. Therefore, this action restricts to an action on the set C of irreducible characters of subgroups of G and this restriction is related with the action of G on CG by conjugation via the formula e(χ · g) = e(χ) · g, for χ ∈ C and g ∈ G. This is easily seen using the formula of Theorem 1.27. Indeed, X 1 g −1 g e(χ · g) = g (χ · g)(1)(χ · g)((h ) )h |H | hg∈Hg 1 X = χ(1)χ(h−1)hg |H| h∈H = g−1e(χ)g = e(χ) · g. Thus, the G-stabilizer of a character χ of a subgroup H of G and the stabilizer of the corresponding idempotent e(χ) coincide and are exactly g Gχ = Ge(χ) = {g ∈ NG(H) | χ(h ) = χ(h), for every h ∈ H}. At last, we define some central element in the group ring, which will play an important role. Given two subgroups H and K of G such that K ¡ H, let e(G, H, K) denote the sum of all G-conjugates of (H,K). Since the G-stabilizer of (H,K) is exactly CenG((H,K)), we get the following formula for T a right transversal of CenG((H,K)) in G: X e(G, H, K) = (H,K)t. t∈T Clearly e(G, H, K) is a central element of QG and if the G-conjugates of (H,K) are orthogonal, then e(G, H, K) is a central idempotent of QG.

2.2 The primitive central idempotent associated to a monomial character

In this section we show that the primitive central idempotent of QG associated to a monomial irreducible complex character can be computed using the elements of the form e(G, H, K).

2.2. The primitive central idempotent associated to a monomial character 30 Chapter 2. Primitive central idempotents of rational group algebras

Definition 2.10. A character χ of G is called monomial if there exist a subgroup H ≤ G and a linear character ψ of H such that χ = ψG, the induced character on G. The group G is called monomial or M-group if all its irreducible characters are monomial.

Theorem 2.11. Let G be a ﬁnite group, H a subgroup of G, χ a linear character of H and χG the induced character of χ on G. If χG is irreducible then the primitive central idempotent of QG associated to χG is

G [CenG((H,K)) : H] eQ(χ ) = e(G, H, K), [Q(χ): Q(χG)] where K is the kernel of χ. Proof. The actions of Aut(C) and G on CG are compatible in the sense that σ · (x · g) = (σ · x) · g, for σ ∈ Aut(C), g ∈ G and x ∈ CG. Thus the notation σ · x · g is unambiguous and one can consider Aut(C) × G acting on the left on CG and C, the set of irreducible characters of subgroups of G, by (σ, g) · x = σ · x · g−1, for σ ∈ Aut(C), g ∈ G and x ∈ CG and (σ, g) · χ = σ · χ · g−1, for σ ∈ Aut(C), g ∈ G and χ ∈ C. Let e = e(χ). The elements of the Aut(C) × G-orbit of the element e can be collected in a table σ1 · e · g1 σ1 · e · g2 ··· σ1 · e · gm σ2 · e · g1 σ2 · e · g2 ··· σ2 · e · gm ············ σn · e · g1 σn · e · g2 ··· σn · e · gm where T = {σ1, . . . , σn} is a left transversal of the Aut(C)-stabilizer of e and TG = {g1, . . . , gm} is a right transversal of the G-stabilizer of e. One may take T = Gal(Q(χ)/Q). By Theorem 2.6, the G-stabilizer of e is Gχ = H, hence TG is a right transversal of H in G. Thus 1 X e(χG) = χG(1)χG(g−1)g |G| g∈G ! X Xm 1 |G| ◦ −1 −1 = χ(1) χ (gig gi ) g |G| g∈G |H| i=1 Xm X 1 −1 −1 = χ(h )gi hgi |H| i=1 h∈H Xm = e · gi. i=1

G Hence, the sum of the elements in the i-th row is σi · e(χ ). By (2.1) and Lemma 2.5, the sum of the elements of the j-th column is (H,K) · gj. We complete the table by adding this

2.2. The primitive central idempotent associated to a monomial character 31 Chapter 2. Primitive central idempotents of rational group algebras sums. G σ1 · e · g1 σ1 · e · g2 ··· σ1 · e · gm σ1 · e(χ ) G σ2 · e · g1 σ2 · e · g2 ··· σ2 · e · gm σ2 · e(χ ) ············ ··· G σn · e · g1 σn · e · g2 ··· σn · e · gm σn · e(χ ) (H,K) · g1 (H,K) · g2 ··· (H,K) · gm ∗ We can compute the total sum ∗ by adding the elements of the last column or the elements of the last row: Xn Xm G ∗ = σi · e(χ ) = (H,K) · gj. (2.3) i=1 j=1

In the ﬁrst sum of (2.3) the elements to add are the elements of the Aut(C)-orbit of e(χG), each of them repeated [Q(χ): Q(χG)] times. Using (2.1), one has X G G G G ∗ = [Q(χ): Q(χ )] σ · e(χ ) = [Q(χ): Q(χ )]eQ(χ ). (2.4) σ∈Gal(Q(χG)/Q) Similarly the second sum of (2.3) adds up the elements of the G-orbit of (H,K), each of them repeated [CenG((H,K)) : H] times. Therefore

∗ = [CenG((H,K)) : H]e(G, H, K). (2.5)

The theorem follows by comparing (2.4) and (2.5). 2

The following two Corollaries follow immediately.

Corollary 2.12. If (H,K) is a Shoda pair of G, then there is an α ∈ Q, necessarily unique, such that αe(G, H, K) is a primitive central idempotent of QG. Proof. This follows immediately, since given a Shoda pair (H,K), we ﬁnd a linear character χ of H with kernel K and, by Proposition 2.8, the induced character χG is irreducible. 2

Corollary 2.13. A ﬁnite group G is monomial if and only if every primitive central idempotent of QG is of the form αe(G, H, K) for α ∈ Q and (H,K) a Shoda pair of G. Proof. The necessarily implication follows immediately from Theorem 2.11, since every G primitive central idempotent of QG is of the form eQ(χ ), for χ a linear character of a subgroup H of G. We prove the other implication. Let χ be an irreducible character of G and take the primitive central idempotent eQ(χ). Then, there exists an α ∈ Q and a Shoda pair (H,K) such that eQ(χ) = αe(G, H, K). By Proposition 2.8, every Shoda pair is associated to a linear character ψ of H with kernel K, such that ψG is irreducible. From the Theorem, we G have that eQ(ψ ) = βe(G, H, K), for a β ∈ Q. Since β is unique such that βe(G, H, K) G is a primitive centralP idempotent, we have that α =Pβ and hence eQ(χ) = eQ(ψ ). Recall G G that e (χ) = e(σ · χ) and e (ψ ) = G e(σ · ψ ). Therefore, it Q σ∈Gal(Q(χ)/Q) P Q σP∈Gal(Q(ψ )/Q) G 1 ◦ −1 1 ◦ −1 G follows that χ = σ · ψ = |H| x∈G σ(ψ (xgx )) = |H| x∈G(σ · ψ) (xgx ) = (σ · ψ) , for a σ ∈ Aut(C). Hence, χ is monomial. 2

2.2. The primitive central idempotent associated to a monomial character 32 Chapter 2. Primitive central idempotents of rational group algebras

So far, we have seen that the primitive central idempotent of QG associated to a monomial irreducible character χG is of the form αe(G, H, K) for α ∈ Q and a Shoda pair (H,K) of G G. In that case we say that the primitive central idempotent eQ(χ ) is realizable by the Shoda pair (H,K). Now we search suﬃcient conditions for α to be 1. Clearly α = 1 if and only if e(G, H, K) is an idempotent. This happens, for example, if the G-conjugates of (H,K) are orthogonal. We say that a subgroup H of G is a maximal abelian subgroup of G if it is maximal in {A ≤ G | A abelian}.

Definition 2.14. A strong Shoda pair of G is a pair (H,K) of subgroups of G satisfying the following conditions:

(SS1) K ≤ H ¡ NG(K),

(SS2) H/K is cyclic and a maximal abelian subgroup of NG(K)/K,

g (SS3) for every g ∈ G \ NG(K), (H,K)(H,K) = 0. We will show that if (H,K) is a strong Shoda pair of G, then it is a Shoda pair of G and then e(G, H, K) is a primitive central idempotent of QG. In this case, we can give an approach to determine a description of the structure of the simple component QGe(G, H, K). We use several times the following obvious facts for every N ≤ K ¡ H ≤ G such that N ¡ G and g ∈ G:

(H,K)g = (Hg,Kg), (2.6)

ωN ((H,K)) = (H/N, K/N). (2.7)

Lemma 2.15. Let K ¡ H ≤ G.

1. If H ¡ NG(K), then NG(K) ≤ CenG((H,K)).

2. If H/K is cyclic then CenG((H,K)) ≤ NG(K) and the following conditions are equivalent for g ∈ G:

(i) g ∈ K, (ii) g(H,K) = (H,K), (iii) ge (H,K) = (H,K).

Proof. The ﬁrst statement is a consequence of (2.6). Indeed, take g ∈ NG(K), then g g g (H,K) = (H ,K ) = (H,K), since H ¡ NG(K). We prove the second statement. Assume that H/K is cyclic. 2(i) implies 2(ii) since gKf = f K, for g ∈ K. The equivalence between 2(ii) and 2(iii) is obvious. Take g ∈ CenG((H,K)) and k ∈ K. Then g−1kg(H,K) = g−1k(H,K)g = g−1(H,K)g = (H,K) and g−1kg ∈ K, as consequence of the equivalence between 2(i) and 2(ii), and hence CenG((H,K)) ≤ NG(K). So we only have to prove that 2(ii) implies 2(i). By Lemma 2.3, one has that (H/K) 6= 0 and so (H,K) 6= 0. Assume that g(H,K) = (H,K). Then supp(g(H,K)) 6= ∅ and g(H,K) = (H,K) ∈ QH, so g ∈ H. Assume

2.2. The primitive central idempotent associated to a monomial character 33 Chapter 2. Primitive central idempotents of rational group algebras that g∈ / K, then hgKi ≤ H/K and let M/K ∈ M(hK, gi/K) ⊆ M(H/K). Therefore M = hK, go(g)/pi, for p a prime divisor of o(g). Moreover hg, Ki = hg, Mi and hence geKf = á á hg, Ki = hg, Mi = geMÝ, since K ¡ H and M ¡ H. Therefore (H,K) = ge (H,K) = 0, a contradiction. Hence g ∈ K. 2

Proposition 2.16. The following conditions are equivalent for a pair (H,K) of subgroups of G: 1. (H,K) is a strong Shoda pair of G,

2. (H,K) is a Shoda pair of G, H¡NG(K) and the G-conjugates of (H,K) are orthogonal.

Moreover, if the previous conditions holds then CenG((H,K)) = NG(K) and e(G, H, K) is a primitive central idempotents. Proof. We first prove that2 implies1. The only things to prove are (SS2) and (SS3). By Lemma 2.15, we have that CenG((H,K)) = NG(K), so (SS3) follows immediately because the G-conjugates are orthogonal. Suppose (SS2) does not hold. Then there exist a subgroup L so that H L NG(K) and H/K L/K NG(K)/K and L/K abelian. Take l ∈ L. Then h−1l−1hl ∈ [H, l] ∩ H, so, because H/K is abelian, h−1l−1hlK = K. Hence, h−1l−1hl ∈ K. Thus, l ∈ H, by (S3). We obtain that H = L, a contradiction. Now we prove that1 implies2. Assume that ( H,K) is a strong Shoda pair. Clearly (H,K) satisfies (S1) and (S2). By Lemma 2.15, CenG((H,K)) = NG(K) and hence condition (SS3) implies that the G-conjugates of (H,K) are orthogonal and so e(G, H, K) is a central idempotent. So the α of Theorem 2.11 must be 1 and therefore e(G, H, K) is a primitive central idempotent. It only remains to show that (H,K) satisfies condition (S3). Let g ∈ G be such that [H, g] ∩ H ⊆ K and χ a linear character of H with kernel K. If h ∈ H and g−1hg ∈ H, then [h, g] ∈ [H, g] ∩ H and thus χ([h, g]) = 1. Therefore 1 X e(χ)e(χ · g−1) = e(χ)(e(χ) · g−1) = χ(h−1)χ(h−1)h gh g−1 |H|2 1 2 1 2 h1,h2∈H and hence, the coefficient of 1 in e(χ)e(χ · g−1) is

X X X g−1 1 −1 −1 1 1 |H ∩ H | 2 χ(h )χ(g hg) = 2 χ([h, g]) = 2 1 = 2 6= 0. |H| h,g−1hg∈H |H| h,g−1hg∈H |H| h,g−1hg∈H |H| P Since (H,K) = σ∈Gal(Q(χ)/Q) e(σ · χ) and the diﬀerent e(σ · χ)’s are orthogonal, we have 0 6= e(χ)e(χ · g−1) = e(χ)(H,K)(H,K)g−1 e(χ · g−1),

g−1 −1 and hence (H,K)(H,K) 6= 0. By condition (SS3), g ∈ NG(K) and so g ∈ NG(K). Then, for all h ∈ H, g−1hg ∈ H, by condition (SS1), and hence [H, g] ⊆ H. Thus [H, g] ⊆ H ∩ [H, g] ⊆ K and hence hH, gi/K is abelian. We deduce that g ∈ H from condition (SS2). This proves (S3). 2

Proposition 2.17. Let (H,K) be a strong Shoda pair and let k = [H : K], N = NG(K), n = [G : N], yK a generator of H/K and φ : N/H → N/K a left inverse of the canonical

2.2. The primitive central idempotent associated to a monomial character 34 Chapter 2. Primitive central idempotents of rational group algebras

σ projection N/K → N/H. Then QGe(G, H, K) is isomorphic to Mn(Q(ξk) ∗τ N/H) and the action and twisting are given by

i φ(nH) i σnH (ξk) = ξk, if yK = y K, 0 j 0 −1 0 j τ(nH, n H) = ξk, if φ(nn K) φ(nK)φ(n K) = y K, for nH, n0H ∈ N/H and integers i and j. Proof. Set f = (H,K), e = e(G, H, K) and T a right transversal for N in G. By Lemma 2.15, we have N = CenG((H,K)). By Proposition 2.16, e is a primitive central idempotent of QG. Remember that kf = f for all k ∈ K and H = hy, Ki. So QHf = Q(yf) is σ1 isomorphic to Q(ξk) via the map given by yf 7→ ξk. Furthermore, QNf = QHf ∗τ1 N/H is a crossed product of N/H over the ﬁeld QHf, with φ(N/H) as a homogeneous basis. Clearly the isomorphism QHf ' Q(ξk) extends naturally to an N/H-graded isomorphism σ1 σ QNf = QHf ∗τ1 N/H ' Q(ξk) ∗τ N/H. If g ∈ G, then the map x 7→ xg is an isomorphism between the QG-modules QGf and g g n QGf . Therefore QGQGe = ⊕g∈T QGf ' (QGf) . By (SS3), we know that ∀g ∈ G \ N fgf = gf gf = 0. Therefore, fQNf = fQGf. Moreover, f is central in QN, so fQGf = QNf. Thus

σ QGe ' EndQG(QGe) ' Mn(EndQG(QGf)) ' Mn(fQGf) = Mn(QNf) ' Mn(Q(ξk)∗τ N/H). 2

σ Remark 2.18. The action σ of the crossed product Q(ξk)∗τ N/H in Theorem 2.17 is faithful. Therefore it can be described as a classical crossed product (Q(ξk)/F, τ), where F is the center of the algebra, which is determined by the Galois action σ.

Although condition (SS3) is very easy to check, it is conceptually disappointing because it has to be checked in QG rather than in the lattice of subgroups of G. The following Corollary shows some suﬃcient conditions to be checked only in the lattice of subgroups of G for e(G, H, K) to be a primitive central idempotent.

Corollary 2.19. Let (H,K) be a pair of subgroups of a ﬁnite group G satisfying the following conditions:

1. K ¡ H ¡ G,

2. H/K is cyclic and a maximal abelian subgroup of NG(K)/K. Then (H,K) is a strong Shoda pair and hence e(G, H, K) is a primitive central idempotent of QG. Proof. Clearly (H,K) satisﬁes (SS1) and (SS2). By condition1 and equation (2.6), the G- conjugates of (H,K) are of the form (H,Kg), with g ∈ G. Since H/Kg is cyclic, Proposition g 2.4 yields that (H,Kg) is a primitive central idempotent of QHKg ' Q(H/Kg), and thus also of QH. So the G-conjugates of (H,K) are primitive central idempotents of QH. Then the G-conjugates of (H,K) are mutually orthogonal and hence (SS3) follows from the equality NG(K) = CenG((H,K)) which is a consequence of Lemma 2.15. 2

2.2. The primitive central idempotent associated to a monomial character 35 Chapter 2. Primitive central idempotents of rational group algebras

2.3 Abelian-by-supersolvable groups

In this section we show that if G is an abelian-by-supersolvable ﬁnite group, then every primitive central idempotent of QG is of the form e(G, H, K) for a strong Shoda pair (H,K). Recall that a group is supersolvable if there is a series of normal subgroups of G with cyclic factors.

Definition 2.20. A group G is abelian-by-supersolvable if there exists an abelian normal subgroup A such that G/A is supersolvable. Notice that the class of abelian-by-supersolvable groups is closed under subgroups and epimorphic images. We need three lemmas. The ﬁrst one is trivial.

Lemma 2.21. Let {K1,K2,...,Km} be an ordered list of non trivial subgroups of a ﬁnite 0 0 0 0 group G satisfying the following condition: if k1k2 ··· km = k1k2 ··· km, with ki, ki ∈ Ki for 0 every i = 1, . . . , m, then ki = ki for all i. Then g g Þ (1 − K1)(1 − K2) ··· (1 − Km) 6= 0.

Lemma 2.22. Let H and K be subgroups of G such that K ¡ H, H/K is cyclic and the G-conjugates of (H,K) are mutually orthogonal, then CenG((H,K)) = NG(K).

Proof. By Lemma 2.15, CenG((H,K)) ≤ NG(K). To prove the converse inclusion, it g is suﬃcient to show that if g ∈ NG(K) then (H,K)(H,K) 6= 0, since the G-conjugates are mutually orthogonal. Let g ∈ NG(K). Since H ≤ NG(K), we have that both (H,K) g g and (H,K) = (H ,K) belong to QNG(K). Therefore we may assume, without loss of generality, that NG(K) = G, that is K ¡ G. Factoring out by K and using the isomorphism f ωK : QGK ' Q(G/K) and equality (2.7), one may assume that K = 1. So H is cyclic and we g have to prove that (H)(H) 6= 0. Let M1,...,Mk be the minimal subgroups of H. Then, g g Mi = Mi or Mi ∩Mi = {1}. Therefore we can order the set of minimal subgroups of H so that g there is an integer h ≤ k such that Mi = Mi if and only if h < i ≤ k. Since Mi ∩ Mj = {1}, if i 6= j, and H is cyclic, we have a direct sum M1M2 ··· Mk = M1 ⊕ M2 ⊕ · · · ⊕ Mk. Since g H is cyclic, each Mi is of an unique prime order and therefore Mi = Mj cannot occur for g g g g i 6= j and hence Mi ∩ Mj = {1}. Therefore M1M2 ··· Mk ∩ M1 M2 ··· Mh = {1} and the list g g of subgroups {M1,...,Mk,M1 ,...,Mh } satisﬁes the conditions of Lemma 2.21. Then

g g g Þg Þg (H)(H) = (1 − M1) ··· (1 − Mk)(1 − M1 ) ··· (1 − Mh ) 6= 0. 2

We say that N is a maximal abelian normal subgroup of G if N is maximal in {A ¡ G | A is abelian}. In general not every maximal abelian normal subgroup of G is a maximal abelian subgroup of G. We given an example where it is not the case.

Example 2.23. The group A5 is simple. So {1} is a maximal abelian normal subgroup of A5. However, {1} is not a maximal abelian subgroup, since for example, h(1, 2)(3, 4)i is an abelian subgroup.

2.3. Abelian-by-supersolvable groups 36 Chapter 2. Primitive central idempotents of rational group algebras

However it is the case when G is supersolvable.

Lemma 2.24. If G is a supersolvable group and N is a maximal abelian normal subgroup of G, then N is a maximal abelian subgroup of G.

Proof. Let K = CenG(N). We have to show that K = N. So assume the opposite. Since G/N is supersolvable and K/N is a normal subgroup of G/N, G/N has a series of normal subgroups of G/N with cyclic factors containing K/N. Write the series as follows

{1} = G0/N ¡ ··· ¡ K/N ¡ ··· ¡ Gn/N = G/N, where possibly K/N = G/N, but K/N 6= G0/N = {1}. Since G1/N is cyclic, there exists x ∈ G1 \ N ⊆ K \ N such that hN, xi/N = G1/N ¡ G/N and so hN, xi ¡ G. Since x ∈ K, hN, xi is an abelian normal subgroup of G containing N properly. This is in contradiction with the maximality of N. 2

Theorem 2.25. Let G be a ﬁnite abelian-by-supersolvable group and e ∈ QG. Then the following conditions are equivalent.

1. e is a primitive central idempotent of QG. 2. e = e(G, H, K) for a strong Shoda pair (H,K) of G.

3. e = e(G, H, K) for a pair (H,K) of subgroups of G satisfying the following conditions:

(A) K ¡ H ¡ CenG((H,K));

(B) H/K is cyclic and a maximal abelian subgroup of CenG((H,K))/K; (C) the G-conjugates of (H,K) are orthogonal.

We say that every primitive central idempotent of QG is realizable by a strong Shoda pair. Proof. To show that2 and3 are equivalent, it is enough to prove that if a pair ( H,K) of subgroups of G satisﬁes either condition2 or condition3, then Cen G((H,K)) = NG(K). That the latter holds in the respective cases follows from Lemma 2.15 and Lemma 2.22. That2 implies1 is a consequence of Proposition 2.16. It remains to show that1 implies3. Let us call a good idempotent to be one of the form e(G, H, K) with (H,K) a pair of subgroups of G satisfying the conditions 3(A)-3(C). Since 3 implies1, every good idempotent is a primitive central idempotent and we want to show that every primitive central idempotent of QG is good or equivalently that 1 is a sum of good idempotents. We argue by induction on |G| with the case |G| = 1 being trivial. Let A be a maximal element in the set of abelian normal subgroups of G such that G/A is supersolvable. If A = {1}, then 1 = e(G, {1}, {1}) and ({1}, {1}) satisﬁes the conditions 3(A)-3(C). Therefore, we may assume that A 6= {1}. Let N be the set of non trivial normal Q f subgroups of G contained in A. Set E = N∈N (1 − N). We are going to obtain our aim by showing that E and 1 − E are both sums of good idempotents. Let e be a primitive central idempotent of QG(1 − E). Then eE = 0 and therefore there is N ∈ N such that eNf 6= 0, otherwise e = eE = 0. Since e is the only central idempotent in f f QG(1 − E)e = QGe, we have e = eN ∈ QGN and ωN (e) is a primitive central idempotent

2.3. Abelian-by-supersolvable groups 37 Chapter 2. Primitive central idempotents of rational group algebras of Q(G/N). By the induction hypothesis ωN (e) = e(G/N, H/N, K/N), where (H/N, K/N) satisﬁes conditions 3(A)-3(C) as a pair of subgroups of G/N. By equations (2.6) and (2.7) and the equality Cen ((H,K)) Cen ((H/N, K/N)) = G , G/N N

ωN (e(G, H, K)) = e(G/N, H/N, K/N). So e = e(G, H, K) and (H,K) satisﬁes conditions 3(A)-3(C) as subgroups of G. This proves that 1 − E is a sum of good idempotents. Let B be the set of subgroups H of A such that A/H is cyclic. By Proposition 2.4, {(A, H) | H ∈ B} is the set of primitive central idempotents of QA. We notice that E is a central idempotent in QA. Therefore E(A, H) = (A, H) or E(A, H) = 0, for all H ∈P B. Let B1 = {H ∈ B | E(A, H)P = (A, H)} and B2 the complementP of B1 in B. Then

1 = H∈B (A, H) and therefore E = H∈B1 (A, H) and 1 − E = H∈B2 (A, H). We claim that B1 = {H ∈ B | N 6⊆ H for every N ∈ N }. Let H ∈ B. If N ⊆ H for some N ∈ N then Nf (A, H) = (A, H), by Lemma 2.15. Therefore E(A, H) = 0 and hence H ∈ B2. Conversely, assume N 6⊆ H for every N ∈ N . By Lemma 2.15, for every N ∈ N there exists n ∈ N such that n(A, H) 6= (A, H) and hence Nf (A, H) 6= (A, H). Then f N(A, H) = 0 and so E(A, H) = (A, H). Thus H ∈ B1. This proves the claim. Since A ¡ G, B1 is closed under conjugation by elements of G and hence E is the sum of the idempotents of the form e(G, A, B) with B running through a set of representatives of the G-conjugates of elements in B1. Therefore we only have to show that e(G, A, B) is a sum of good idempotents for every B ∈ B1. In the remainder of the proof, B is an element of B1. If B = 1 then A is cyclic and hence G is supersolvable. By Lemma 2.24, A is a maximal abelian subgroup of G. Obviously CenG((A, 1)) = G and hence the pair (A, 1) satisﬁes conditions 3(A)-3(C), so that e(G, A, 1) is a good idempotent. Assume now that B 6= 1. Thus B is a non trivial subgroup of G which does not contain any non trivial normal subgroup of G and in particular B is not normal in G, that is NG(B) 6= G. Let S = CenG((A, B)). By Lemma 2.15, S = NG(B). Since (A, B) is a central idempotent of QS and S is a proper subgroup of G, by the induction hypothesis Xk (A, B) = e(S,Hi,Ki), i=1 where each (Hi,Ki) is a pair of subgroups of S satisfying conditions 3(A)-3(C) as subgroups of S. g We claim that if g ∈ G \ S, then (Hi,Ki)(Hi,Ki) = 0. This is because (A, B) and (A, B)g are two diﬀerent primitive central idempotents of QA and hence g g g (Hi,Ki)(Hi,Ki) = (Hi,Ki)(A, B)(A, B) (Hi,Ki) = 0.

From the claim it follows that CenG((Hi,Ki)) ⊆ S and therefore

CenG((Hi,Ki)) = CenS((Hi,Ki)). (2.8)

Let T be a right transversal of S in G and, for each i ∈ {1, . . . , k}, let Ri be a right transversal of CenS((Hi,Ki)) in S. By (2.8), RiT is a right transversal for CenG((Hi,Ki)) in G. So

X X Xk X Xk X Xk t t rt e(G, A, B) = (A, B) = e(S,Hi,Ki) = (Hi,Ki) = e(G, Hi,Ki). t∈T t∈T i=1 t∈T i=1 r∈Ri i=1

2.3. Abelian-by-supersolvable groups 38 Chapter 2. Primitive central idempotents of rational group algebras

Recall that the (Hi,Ki) satisfy conditions 3(A)-3(C) as subgroups of S. Now we show that they also satisfy these conditions as subgroups of G. 3(A) and 3(B) follow immediately from (2.8) and 3(C) is a consequence of the claim. Thus e(G, A, B) is a sum of good idempotents and this ﬁnishes the proof. 2

As a direct consequence one has:

Corollary 2.26. Let G be a ﬁnite supersolvable group and e ∈ QG. Then the following conditions are equivalent.

1. e is a primitive central idempotent of QG. 2. e = e(G, H, K) for a pair (H,K) of subgroups of G satisfying the following conditions:

(a) K ≤ H ≤ NG(K);

(b) H/K is cyclic and a maximal abelian normal subgroup of NG(K)/K; g (c) if g ∈ G \ NG(K), then (H,K)(H,K) = 0. 3. e = e(G, H, K) for a pair (H,K) of subgroups of G satisfying the following conditions:

(A) K ¡ H ≤ CenG((H,K));

(B) H/K is cyclic and a maximal abelian normal subgroup of CenG((H,K))/K; g (C) if g ∈ G \ CenG((H,K)), then (H,K)(H,K) = 0. Note that conditions 2(a)-2(c) are stronger than saying that (H,K) is a strong Shoda pair, since H/K has to be a maximal abelian normal subgroup of NG(K)/K. However we proved in Lemma 2.24, that in supersolvable groups a maximal abelian normal subgroup is a maximal abelian subgroup. By Corollary 2.13 we know that every abelian-by-supersolvable group is monomial. Note that we have not used this fact in the proof of Theorem 2.25. In fact one can deduce a stronger result from Theorem 2.11 and Theorem 2.25.

Corollary 2.27. If G is an abelian-by-supersolvable ﬁnite group then every irreducible character of G is a monomial character induced by a linear character χ of a subgroup H of

G such that H is normal in NG(ker χ) and the elements of the G-orbit of eQ(χ) are mutually orthogonal.

Proof. Let ψ be an irreducible character of G. By Theorem 2.25, eQ(ψ) = e(G, H, K), for a strong Shoda pair (H,K). Let χ be the linear character of H with kernel K, associated to the G G Shoda pair (H,K). Then χ is irreducible. Now consider eQ(χ ). By Theorem 2.11, there G exists α ∈ Q such that eQ(χ ) = αe(G, H, K). Because e(G, H, K) is already an idempotent, G necessarily, α = 1, and hence, eQ(ψ) = eQ(χ ). Therefore, there exists σ ∈ Aut(C) such G G that ψ = σ · χ = (σ · χ) , i.e. ψ is monomial. Moreover H is normal in NG(K) and the

G-conjugates of (H,K) are orthogonal. Since, by Lemma 2.5, eQ(σ · χ) = (H,K), the statement follows. 2

This Corollary brings us to the following Deﬁnition.

2.3. Abelian-by-supersolvable groups 39 Chapter 2. Primitive central idempotents of rational group algebras

Definition 2.28. Let G be a ﬁnite group and χ an irreducible character of G. One says that χ is strong monomial if there is a strong Shoda pair (H,K) of G and a linear character ψ of H with kernel K such that χ = ψG, the induced character on G. The group G is strong monomial if every irreducible character of G is strong monomial. Then we can restate Corollary 2.27 as follows.

Corollary 2.29. An abelian-by-supersolvable ﬁnite group is strong monomial. Now we show how to modify the proof of Theorem 2.25 to show that the primitive central idempotents provided by Corollary 2.19 are enough to describe the primitive central idempotents of QG for a metabelian group. Recall that a group G is metabelian if it contains an abelian normal subgroup A so that G/A is abelian too or equivalently if G0 is abelian.

Theorem 2.30. Let G be a metabelian ﬁnite group and let A be a maximal abelian subgroup of G containing G0. The primitive central idempotents of QG are the elements of the form e(G, H, K), where (H,K) is a pair of subgroups of G satisfying the following conditions:

1. H is a maximal element in the set {B ≤ G | A ≤ B and B0 ≤ K ≤ B};

2. H/K is cyclic.

Proof. Let G and A be as in the statement of the theorem. Note that every subgroup H of G containing A is normal in G. Moreover, if K ≤ B ≤ G, then B0 ≤ K if and only if B ⊆ NG(K) and B/K is abelian. Thus if the pair (H,K) satisfies conditions1 and2, then it also satisfies conditions1 and2 of Corollary 2.19 and so, e = e(G, H, K) is a primitive central idempotent of QG. Now we want to prove that every primitive central idempotent is of this form. Note that this is equivalent to prove that 1 is a sum of primitive central idempotents of the desired form. We argue by induction on the order of G, the case |G| = 1 is trivial. If G = A, then G is abelian and the statement holds by Proposition 2.4. Therefore one may assume that G 6= A. Let B be the set of subgroups of G containing A properly. Note that 0 every element B ∈ B is normal in G and hence so is B . Also by maximality ofQA as abelian 0 0 Ý0 subgroup of G containing G , B is not abelian and hence B 6= {1}. Let E = B∈B(1 − B ), which is a central idempotent of QG. We will show that both E and 1 − E are a sum of e(G, H, K)’s with K and H satisfying conditions1 and2. Let e be a primitive central idempotent of QG(1 − E). Then there is B ∈ B such that Ý0 Ý0 Ý0 eB 6= 0, for otherwise 0 = eE = e. Thus eB = e ∈ QGB and ωB0 (e) is a primitive 0 0 0 0 central idempotent of Q(G/B ). By the induction hypothesis ωB0 (e) = e(G/B , H/B , K/B ), where (H/B0, K/B0) satisfies conditions1 and2. Then e = e(G, H, K) and (H,K) satisfies conditions1 and2. This proves that 1 − E is a sum of primitive central idempotents of the desired form. LetP H be the set of subgroups H of A such that A/H is cyclic. By Proposition 2.4, 1 = H∈H (A, H). Also, E is a central idempotent in QA and therefore, (A, H)E = 0 or (A, H)E = (A, H), for H ∈ H. LetP H1 = {H ∈ H | (A, H)PE = (A, H)} and H2 the complement of H1 in H. Then E = H∈H1 (A, H) and 1 − E = H∈H2 (A, H). 0 0 We claim that H1 = {H ∈ H | B 6⊆ H for every B ∈ B}. Let H ∈ H. If B ⊆ H for some B ∈ B, then BÝ0(A, H) = (A, H), by Lemma 2.15. Therefore, (A, H)E = 0 and hence

2.3. Abelian-by-supersolvable groups 40 Chapter 2. Primitive central idempotents of rational group algebras

0 H ∈ H2. Conversely, assume that B 6⊆ H for every B ∈ B. By Lemma 2.15, for every B ∈ B, there exists b ∈ B0 such that b(A, H) 6= (A, H) and hence BÝ0(A, H) 6= (A, H). Ý0 Then B (A, H) = 0 and thus (A, H)E = (A, H). Thus H ∈ H1. This proves the claim. 0 By the claim, if H ∈ H1, then A is maximal in the set of subgroups B of G such that B ⊆ H, that is (A, H) satisﬁes conditions1 and2. Consider GPacting on H1 by conjugationP and let

R be a set of representatives of this action. Then E = H∈H1 (A, H) = H∈R e(G, A, H). This ﬁnishes the proof. 2

Often when the primitive central idempotents of QG are sought, one recalls that the primi- f tive central idempotents of QGG0 ' Q(G/G0) are easy to compute by using the description of the primitive central idempotents of rational groups algebras of abelian groups, see Proposi- f tion 2.4, and one concentrates on computing the primitive central idempotents of QG(1−G0). Now we can go further. Indeed, for every group G, the quotient group G/G00 is metabelian Ý and using the isomorphism QGG00 ' Q(G/G00) one deduces the following from Theorem 2.30.

Corollary 2.31. Let G be a ﬁnite group and A a maximal element in {K ≤ G | G0 ≤ Ý K and K/G00 is abelian}. Then every primitive central idempotent of QGG00 is of the form e(G, H, K) for a pair (H,K) of subgroups of G satisfying the following conditions:

1. G00 ≤ K;

2. H is a maximal element in the set {B ≤ G | A ≤ B and B0 ≤ K ≤ B};

3. H/K is cyclic.

2.4 Example: metacyclic groups

We show a straightforward method to compute the primitive central idempotents of QG for a ﬁnite metacyclic group. We also compute the Wedderburn decomposition.

Definition 2.32. A metacyclic group is a group G having a normal cyclic subgroup N = hai such that G/N = hbNi is cyclic. It is well known and easy to verify that a ﬁnite metacyclic group G has a representation of the form [CR88] G = ha, b | am = 1, bn = at, bab−1 = ari, where m, n, t, r are integers satisfying the conditions gcd(r, m) = 1 and m | t(r − 1). Deﬁne σ ∈ Aut(hai) as σ(a) = bab−1 = ar. Hence, σk(a) = ark = bkab−k. Let u be the order of σ, then u | n since a = bnab−n = arn = σn(a). Moreover m | rn − 1. d d −d −1 rd−1 0 rd−1 For every d | u, let Gd = ha, b i. Since b ab a = a ∈ Gd and Gd/ha i is abelian, 0 rd−1 we know that Gd = ha i. Hence Gu is a maximal abelian normal subgroup of G containing G0. By Theorem 2.30, the primitive central idempotents of QG are the elements of the form e(G, Gd,K) where d is a divisor of u and K is a subgroup of Gd satisfying the following conditions:

1. d = min{x | u : arx−1 ∈ K},

2.4. Example: metacyclic groups 41 Chapter 2. Primitive central idempotents of rational group algebras

2. Gd/K is cyclic. If n is a prime number, then there are only two kinds of idempotents: those of the form e(G, G, K) = (G, K) with G/K cyclic and those of the form e(G, hai, hadi) = (hai, hadi) where d is a divisor of m which does not divide r − 1. In this case, the simple components of the Wedderburn decomposition are given by

QGe(G, G, K) ' Q(ξ[G:K]) and d σ QGe(G, hai, ha i) ' Q(ξd) ∗τ hbhaii, with τ and σ as in Proposition 2.17.

2.5 An algorithm to compute the Wedderburn decomposition

In this section, we explain algorithms to compute the primitive central idempotents and the Wedderburn decomposition of QG for many groups, including abelian-by-supersolvable groups, following [OAdR03]. Olivieri and del R´ıoimplemented this algorithms in the package “Wedderga” for the system GAP [GAP08]. Nowadays, the package has been updated and can compute the Wedderburn decomposition of QG for all finite groups. We will not explain this in detail, more information can be found in [CKO+09]. By Proposition 2.16 the rule (H,K) → e(G, H, K) defines a map eG from the set SSP of strong Shoda pairs of G to the set E of primitive central idempotents of QG. The image is the set ESSP of primitive central idempotents of QG realizable by strong Shoda pairs of QG. We say that a subset S of SSP is complete and not redundant if eG restricts to a bijection between S and ESSP . Analogously, we can define when a subset S of SP is complete and not redundant, with SP the set of Shoda pairs of G. We will consider 5 algorithms:

• ShodaPairsAndIdempotents(QG): computes a complete and not redundant set of Shoda pairs of G and the primitive central idempotents of QG realizable by those Shoda pairs of G;

• StrongShodaPairsAndIdempotents(QG): computes a complete and not redundant set of strong Shoda pairs of G and the primitive central idempotents of QG realizable by those strong Shoda pairs of G;

• StrongShodaPairs(G): computes a complete and not redundant set of strong Shoda pairs of G;

• SimpleAlgebraByStrongSPInfo(QG, H, K): computes a list of data describing the simple algebra QGe(G, H, K) for a strong Shoda pair (H,K) of G; • SimpleFactorsFromSSP(G): computes a list formed by the output of SimpleAlgebraByStrongSPInfo(QG, H, K) for (H,K) running through a complete and not redundant set of strong Shoda pairs of G.

2.5. An algorithm to compute the Wedderburn decomposition 42 Chapter 2. Primitive central idempotents of rational group algebras

By Corollary 2.13, ShodaPairsAndIdempotents(QG) computes all primitive central idempotents of QG if and only if G is monomial. By Theorem 2.25, if G is abelian-by-supersolvable then E = ESSP and StrongShodaPairsAndIdempotents(QG) computes all the primitive central idempotents of QG. The algorithms ShodaPairsAndIdempotents, StrongShodaPairsAndIdempotents and StrongShodaPairs are similar. We will only explain StrongShodaPairsAndIdempotents, the others can be found in [OAdR03]. Even more, the algorithm we will explain will only return the primitive central idempotents, however, as stated above, the version implemented in GAP also returns the Shoda pairs. Algorithm1 shows the algorithm StrongShodaPairsAndIdempotents(QG) without any detail.

Algorithm 1 StrongShodaPairsAndIdempotents(QG) Require: A ﬁnite group G. 1: C := Conjugacy classes of subgroups of G in decreasing order; 2: Id := ∅; 3: SumId := 0; 4: while SumId 6= 1 do 5: for C ∈ C do 6: K := Representative of C; 7: Search all H such that (H,K) is SSP of G 8: e := e(G, H, K) 9: if SumId · e = 0 then 10: Add e to the list Id; 11: SumId := SumId + e; 12: end if 13: end for 14: end while 15: return Id.

The main loop of Algorithm1 selects one representative K of each conjugacy class C of subgroups of G and searches for H such that (H,K) is a strong Shoda pair. The reason for considering only one representative in each conjugacy class is that if g ∈ G and (H,K) is a strong Shoda pair of G then so is (Hg,Kg) and e(G, Hg,Kg) = e(G, H, K). At a certain moment, the list Id contains the primitive central idempotents found so far and SumId is the sum of the elements in Id. When a new strong Shoda pair (H,K) is found, we compute e = e(G, H, K) and if e is not in the list Id, then Id is enlarged with e and the value of SumId is changed. The algorithm stops when either the sum SumId = 1 or when all conjugacy classes have been considered. Notice that the conjugacy classes are selected in decreasing order. The reason for this is that some of the “unavoidable” strong Shoda pairs are those of the form (G, K) with G0 ≤ K. This is justiﬁed by the following Proposition. Notice that those subgroups K are somehow “large”. Since we want the algorithm to be as fast as possible, it is desirable that we discover these pairs as soon as possible.

2.5. An algorithm to compute the Wedderburn decomposition 43 Chapter 2. Primitive central idempotents of rational group algebras

Proposition 2.33. 1. If G0 ≤ K ≤ G, then (H,K) is a Shoda pair of G if and only if H = G and G/K is cyclic. In that case (G, K) is a strong Shoda pair and e(G, G, K) = (G, K). f 2. Conversely, if e is a primitive central idempotent of QGG0 then there is a unique strong Shoda pair that realizes e and this strong Shoda pair is of the form (G, K) with G0 ≤ K. Proof. 1. Suppose G0 ≤ K ≤ G and (H,K) is a Shoda pair. By (S3), G = H. Together with Proposition 2.16, this proves the ﬁrst statement. f0 2. Let e be a primitive central idempotents of QGG . Using the isomorphism ωG0 : f0 0 0 QGG ' Q(G/G ), the fact that G/G is abelian and Proposition 2.4, we obtain that ωG0 (e) = (G/G0, K/G0) for a subgroup K of G such that G/K is cyclic. Using equation (2.7), we know that e = (G, K) with G0 ≤ K and G/K cyclic, that is (G, K) is a strong Shoda pair. It remains to show the uniqueness of the strong Shoda pair. Let (H1,K1) be a strong Shoda pair of G such that e = e(G, H1,K1). By (SS3), one has f0 f0 0 that G (H1,K1) = G e(H1,K1) = e(H1,K1) = (H1,K1). Then for every g ∈ G , we have f0 f0 g(H1,K1) = gG (H1,K1) = G (H1,K1) = (H1,K1), and by Lemma 2.15, we have that 0 G ≤ K1. By1, H1 = G and (G, K) = e = (G, K1). Using again Lemma 2.15, one deduces that K = K1. 2

Thus, if G0 ≤ K ≤ G, we only have to check whether (G, K) is a strong Shoda pair or not. However, if G0 6⊆ K ≤ G, there might be more than one candidate for H to form a strong Shoda pair (H,K). This is shown in the following example.

4 2 2 y −1 Example 2.34. If Q8 = hx, y | x = x y = 1, x = x i is the quaternion group of order 8, then it is easy to check that both (hxi, 1) and (hyi, 1) are strong Shoda pairs of Q8. Now we want some help to eliminate subgroups K which cannot appear as the second component of a strong Shoda pair and to reduce the number of subgroups to consider as the ﬁrst component. The following Proposition will help us.

Proposition 2.35. If (H,K) is a strong Shoda pair and N = NG(K) then the following conditions hold:

1. N/H is isomorphic to a subgroup of the group of units of Z/[H : K]Z and in particular N/H is abelian.

2. h(N/K)0,Z(N/K)i ≤ H/K and therefore h(N/K)0,Z(N/K)i is cyclic.

Proof. 1. Consider N/K acting by conjugation on H/K. Obviously, H/K is a subset of N/K the kernel of this action and by (SS2), H/K is the kernel. Hence H/K ' N/H is isomorphic to a subgroup of Aut(H/K). Since H/K is cyclic, it is isomorphic to Z/[H : K]Z and it is an easy exercise that Aut(Z/[H : K]Z) 'U(Z/[H : K]Z). 2. By1, N 0 ≤ H and therefore (N/K)0 = N 0K/K ⊆ H/K. Moreover, Z(N/K)(H/K) is abelian and hence Z(N/K) ⊆ H/K by (SS2). Thus, h(N/K)0,Z(N/K)i ≤ H/K and since H/K is cyclic, so is h(N/K)0,Z(N/K)i. 2

2.5. An algorithm to compute the Wedderburn decomposition 44 Chapter 2. Primitive central idempotents of rational group algebras

Let K be the subgroup selected in the algorithm. Set

0 N = NG(K) and L = h(N/K) ,Z(N/K)i.

By Proposition 2.35, if L is not cyclic, then K cannot appear as the second component of a strong Shoda pair and so, we can pass to the next conjugacy class. Assume that L is cyclic. We are looking for subgroups H of G such that (H,K) is a strong Shoda pair of G. Let Cen = CenN/K (L). By Proposition 2.35, if (H,K) is a strong Shoda pair of G, then L ≤ H/K and since H/K is cyclic, one has that H/K ⊆ Cen. Thus we look for H among the subgroups such that H/K is a cyclic subgroup of Cen containing L. Note that this condition ensures that (H,K) satisﬁes (SS1) (since (N/K)0 ⊆ H/K) and H/K is abelian. Now we have 2 cases, either Cen is abelian, or Cen is not abelian. The ﬁrst case is represented in Algorithm2.

Algorithm 2 part of StrongShodaPairsAndIdempotents(QG) 0 Require: A subgroup K, N := NG(K), L := h(N/K) ,Z(N/K)i is cyclic.

1: Cen := CenN/K (L); 2: if Cen is abelian then 3: H := preimage of Cen in N; 4: if (H,K) satisﬁes (SS2) and (SS3) then 5: e := e(G, H, K) 6: if SumId · e = 0 then 7: Add e to the list Id; 8: SumId := SumId + e; 9: end if 10: end if 11: next C; 12: end if

Assume that Cen is abelian and (H,K) is a strong Shoda pair of G, then H/K = Cen by (SS2). That is why the algorithm only considers the preimage of Cen in N as a candidate for H and passes to the next conjugacy class. Now assume that Cen is not abelian. This case can be found in Algorithm3. When Cen is not abelian, then L is properly contained in H/K by (SS2). Thus we look for cyclic subgroups of Cen containing L properly. More specifically, we consider as a candidate for K, the preimage of hL, xi in N for x ∈ X, where X is initially Cen \ L. Since H/K is abelian, if (H,K) does not satisfy (SS2) then H does not contain any other group H1 such that (H1,K) satisfy (SS2). That is why X is resettled as X \ H/K in that case. We keep looking for H until a pair (H,K) satisfying (SS2) (and (SS1)) is found or X = ∅. Assume we have found a (H,K) satisfying (SS2), then we check if (H,K) satisfies condition (SS3). If it does, then e(G, H, K) is considered as a candidate to enlarge the list Id. Then whether (H,K) satisfies (SS3) or not, we pass to the next conjugacy class. One could argue that maybe there is a different possibility for H to consider. In fact, this H may exist as we have seen in Example 2.34, but we do not want to find all strong Shoda pairs but the primitive central idempotents realizable by strong Shoda pairs. The next Proposition shows that if a pair

2.5. An algorithm to compute the Wedderburn decomposition 45 Chapter 2. Primitive central idempotents of rational group algebras

Algorithm 3 part of StrongShodaPairsAndIdempotents(QG) 0 Require: A subgroup K, N := NG(K), L := h(N/K) ,Z(N/K)i is cyclic.

1: Cen := CenN/K (L); 2: if Cen is not abelian then 3: X := Cen \ L; 4: while X 6= ∅ do 5: select x ∈ X; 6: H/K := hL, xi; 7: H := preimage of H/K in N; 8: if (H,K) satisﬁes (SS2) then 9: if (H,K) satisﬁes (SS3) then 10: e := e(G, H, K) 11: if SumId · e = 0 then 12: add e to the list Id; 13: SumId := SumId + e; 14: end if 15: end if 16: next C; 17: end if 18: X:=X \ H/K; 19: end while 20: end if

(H,K) of subgroups of G satisﬁes conditions (SS1) and (SS2) then e(G, H, K) is determined by K. This justiﬁes us passing to the next conjugacy class.

Proposition 2.36. If (H1,K) and (H2,K) are two pairs of subgroups of G satisfying conditions (SS1) and (SS2) then

1. (H1,K) = (H2,K),

2. e(G, H1,K) = e(G, H2,K),

3. (H1,K) is a strong Shoda pair if and only if (H2,K) is a strong Shoda pair. Proof. 2 and3 are consequences of1, so we only prove1. Since (Hi,K) belongs to QHi ≤ QNG(K) for i = 1, 2, we may assume that NG(K) = G, f f that is K¡G. Note also that (Hi,K) ∈ QGK. Using the isomorphism ωK : QGK ' Q(G/K) and factoring out by K, one may assume that K = {1}. In other words one may assume that H1 and H2 are cyclic, normal and maximal abelian subgroups of G. We prove that every minimal subgroup of H2 is embedded in H1. By symmetry M(H1, 1) = M(H2, 1) and then the proposition follows. A minimal subgroups of H2 is of the form hyi with y of prime order. Let x be a generator −1 j −1 j−1 of H1. Since H1 ¡G, y xy = x , for some j. Then xyx = yx . Since H2 is a cyclic normal j−1 j−1 subgroup of G, so is hyi and hence yx ∈ hyi. Thus x ∈ hyi ∩ H1. If hyi ∩ H1 = hyi, then j−1 y ∈ H1 as desired. Otherwise hyi ∩ H1 = {1} and hence x = 1. Then hH1, yi is abelian and by the maximality of H1, hyi ⊆ H1. 2

2.5. An algorithm to compute the Wedderburn decomposition 46 Chapter 2. Primitive central idempotents of rational group algebras

Algorithm 4 StrongShodaPairsAndIdempotents(QG) Require: A finite group G. 1: C := Conjugacy classes of subgroups of G in decreasing order; 2: Id := ∅; 3: SumId := 0; 4: while SumId 6= 1 do 5: for C ∈ C do 6: K := Representative of C; 7: N := NG(K); 8: L := h(N/K)0,Z(N/K)i; 9: if L is cyclic then 10: Cen := CenN/K (L); 11: if Cen is abelian then 12: H := preimage of Cen in N; 13: if (H,K) satisfies (SS2) and (SS3) then 14: e := e(G, H, K) 15: if SumId · e = 0 then 16: Add e to the list Id; 17: SumId := SumId + e; 18: end if 19: end if 20: next C; 21: else 22: X := Cen \ L; 23: while X 6= ∅ do 24: select x ∈ X; 25: H/K := hL, xi; 26: H := preimage of H/K in N; 27: if (H,K) satisfies (SS2) then 28: if (H,K) satisfies (SS3) then 29: e := e(G, H, K) 30: if SumId · e = 0 then 31: add e to the list Id; 32: SumId := SumId + e; 33: end if 34: end if 35: next C; 36: end if 37: X:=X \ H/K; 38: end while 39: end if 40: end if 41: end for 42: end while 43: return Id.

2.5. An algorithm to compute the Wedderburn decomposition 47 Chapter 2. Primitive central idempotents of rational group algebras

Algorithm4 is the full detailed algorithm of StrongShodaPairsAndIdempotents(QG). We will now explain the algorithms to describe the simple factors using the primitive central idempotents, namely SimpleAlgebraByStrongSPInfo and SimpleFactorsFromSSP. First, consider a general crossed product R ∗ A over a ﬁnite abelian group A. Let A = C1 × · · · × Cm, where each Ci is a cyclic group of order oi, generated by ci. Let gi be an i1 im invertible element of degree ci of R∗A. Then B = {g1 ··· gm : 0 ≤ ij < oj} is a homogeneous basis of R ∗ A. For every i ∈ {1, . . . , m} let:

−1 • αi be the automorphism of R given by αi(r) = gi rgi (r ∈ R),

oi • gi = si ∈ R, −1 −1 • for every i < j ≤ m let τj,i = gj gi gjgi. Then the action and twisting of R ∗ A associated to the homogeneous basis B is determined by the data (oi, αi, si)1≤i≤m and (τj,i)1≤i

im i1 α i1 im = αm ◦ · · · ◦ α1 c1 ···cm

oi and to know τ i1 im j1 jm we only have to know τj,i for i < j. By the equations, gi = si, we c1 ···cm ,c1 ···cm obtain a ﬁnite basis. This can be expressed alternatively by saying that the crossed product R ∗ A is the R-algebra given by the following presentation:

oi R(g1, . . . , gm | rgi = giαi(r), gi = si, gjgi = gigjτj,i, r ∈ R, 1 ≤ i < j ≤ m). (2.9)

Let (H,K) be a strong Shoda pair and let k = [H : K], N = NG(K), n = [G : N], yK a generator of H/K and φ : N/H → N/K a left inverse of the canonical projection N/K → N/H. In Proposition 2.17 we have seen that QGe(G, H, K) ' Mn(Q(ξk) ∗ N/H), with homogeneous basis φ(N/H). From Proposition 2.35 we know that N/H is abelian. We can use these facts to describe the crossed product Q(ξk) ∗ N/H by a list of nonnegative numbers. Say N/H = C1 × · · · × Cm, where each Ci is a cyclic group of order oi, generated by ci. Let gi = φ(ci). Now we have to describe the data oi, αi, si and τj,i to obtain the presentation as in (2.9). Since αi is an automorphism of the cyclotomic ﬁeld Q(ξk), we only need to βi gi βi know αi(ξk) to describe it. By Proposition 2.17, αi(ξk) = ξk if and only if (yK) = (yK) . γi Using again Proposition 2.17 and the isomorphism QHf ' Q(ξk), one has that si = ξk if oi γi δij δij gi = (yK) , and τj,i = ξk if [gj, gi] = (yK) . Therefore QGe(G, H, K) can be described from the following 4-tuple:

(n, k, (oi, βi, γi)1≤i≤m, (δij)1≤i

Proposition 2.37. Let (H,K) be a strong Shoda pair of G and let k = [H : K],N = NG(K) and n = [G : N], yK a generator of H/K and N/H = C1 ×· · ·×Cm, where each Ci is a cyclic group of order oi. For every i ∈ {1, . . . , m}, let gi ∈ N/K be a representative of a generator of Ci. Let (βi)1≤i≤m, (γi)1≤i≤m and (δij)1≤i

2.5. An algorithm to compute the Wedderburn decomposition 48 Chapter 2. Primitive central idempotents of rational group algebras

Then QGe(G, H, K) ' Mn(A) where A is the algebra deﬁned by the following presentation:

βi oi γi δij A = Q(ξk)(g1, . . . , gm | ξkgi = giξk , gi = ξk , gjgi = gigjξk , 1 ≤ i < j ≤ m). (2.11)

The function SimpleAlgebraByStrongSPInfo(QG, H, K) returns a numerical description of the simple algebra QGe(G, H, K), for (H,K) a strong Shoda pair of G. This description is based on tuple (2.10). The output is either a 2-tuple, or a 5-tuple. In the 2-tuple case, the output will be of the form

[n, F ], where n is a positive integer and F is a ﬁeld. Actually, this tuple is equal to the 4-tuple (n, k, ∅, ∅) of (2.10) and hence it represents the n×n-matrix algebra Mn(Q(ξk)) (2.11). Notice that GAP already computes F = Q(ξk). In the 5-tuple case, the output is of the form

[n, F, k, [oi, βi, γi]1≤i≤m, [δij]1≤i

βi oi γi δij A = Q(ξk)(g1, . . . , gm | ξkgi = giξk , gi = ξk , gjgi = gigjξk , 1 ≤ i < j ≤ m). Finally the function SimpleFactorsFromSSP(G) applies SimpleAlgebraByStrongSPInfo to (G, H, K) for (H,K) running through StrongShodaPairs(G). In particular, if all the primitive central idempotents of QG are realizable by strong Shoda pairs, then this function describes the Wedderburn decomposition of QG.

2.6 More examples

At this moment, we can give plenty of examples, using GAP [GAP08] and the Wedderga package [CKO+09]. We will provide some GAP code to illustrate the possibilities. Some functions I wrote myself, can be found in Chapter5. We start with an easy example to compare the new method with the classical method. This is, ﬁrst compute the irreducible characters of G, then compute the primitive central idempotents associated to those characters and ﬁnally compute the simple factors associated to those idempotents.

4 2 Example 2.38. Consider the quaternion group of order 8, Q8 = hx, y | x = 1, x = y2, y−1xy = x−1i. Note that this group is abelian-by-supersolvable. This means that all primitive central idempotents are realizable by strong Shoda pairs and therefore we compute all strong Shoda pairs and their associated idempotents. We also compute the simple algebra associated to the pairs. gap> F:=FreeGroup("x","y");; x:=F.1;;y:=F.2;; gap> G:=F/[x^4,x^2*y^2,y^(-1)*x*y*x];; gap> QG:=GroupRing(Rationals,G);; gap> rc:=StrongShodaPairsAndIdempotents(QG);

2.6. More examples 49 Chapter 2. Primitive central idempotents of rational group algebras rec( StrongShodaPairs := [ [ Q8, Q8 ], [ Q8, Group([ x^-2, y*x^-1 ]) ], [ Q8, Group([ x^-2, y ]) ], [ Q8, Group([ x ]) ], [ Group([ x ]), Group([ ]) ] ], PrimitiveCentralIdempotents := [ (1/8)*+(1/8)*x+(1/8)*y+(1/ 8)*x^2+(1/8)*x*y+(1/8)*x^3+(1/8)*x^2*y+(1/8)*x^3*y, (1/8)*x^-4+(-1/8)*x^-3+(-1/8)*x^-1*y*x^-1+(1/8)*x^-2+(1/8)*x^-4*y*x^ -1+(-1/8)*x^-1+(-1/8)*x^-3*y*x^-1+(1/8)*x^-2*y*x^-1, (1/8)*x^-4+(-1/8)*x^-3+(1/8)*x^-4*y+(1/8)*x^-2+(-1/8)*x^-3*y+(-1/ 8)*x^-1+(1/8)*x^-2*y+(-1/8)*x^-1*y, (1/8)*y^-2*x^2+(1/8)*y^-2*x^3+(-1/8)*y^-1*x^2+(1/8)*y^-2+(-1/8)*y^ -1*x+(1/8)*y^-2*x+(-1/8)*y^-1+(-1/8)*y^-1*x^3, (1/2)*x^4+(-1/2)*x^2 ] ) gap> SimpleAlgebraByStrongSPInfo(QG,rc.("StrongShodaPairs")[1][1], > rc.("StrongShodaPairs")[1][2]); [ 1, Rationals ] gap> SimpleAlgebraByStrongSPInfo(QG,rc.("StrongShodaPairs")[2][1], > rc.("StrongShodaPairs")[2][2]); [ 1, Rationals ] gap> SimpleAlgebraByStrongSPInfo(QG,rc.("StrongShodaPairs")[3][1], > rc.("StrongShodaPairs")[3][2]); [ 1, Rationals ] gap> SimpleAlgebraByStrongSPInfo(QG,rc.("StrongShodaPairs")[4][1], > rc.("StrongShodaPairs")[4][2]); [ 1, Rationals ] gap> SimpleAlgebraByStrongSPInfo(QG,rc.("StrongShodaPairs")[5][1], > rc.("StrongShodaPairs")[5][2]); [ 1, Rationals, 4, [ [ 2, 3, 2 ] ], [ ] ] Notice that the tuples in the output are of one of the following two forms:

[1, Q] [1, Q, 4, [[2, 3, 2]], []]. Using Proposition 2.37, we can describe the simple factors. In the ﬁrst case the list corresponds −1 2 2 to the algebra Q. In the other case, it corresponds to A = Q(ξ4)(g | ξ4g = gξ4 , g = ξ4 = −1). We discover that

2 2 QQ8 = QQ8(Q8,Q8) ⊕ QQ8(Q8, hx , xyi) ⊕ QQ8(Q8, hx , yi) ⊕QQ8(Q8, hxi) ⊕ QQ8(hxi, 1) ' Q ⊕ Q ⊕ Q ⊕ Q ⊕ A.

−1 Notice that the center of A is Q(ξ4 +ξ4 ) = Q and A has the following presentation as algebra over its center: 2 2 −1, −1 A = Q(g, i | g = i = −1, ig = −gi) = = H(Q). Q Now we want to know to which irreducible character each Shoda pair corresponds. There- fore, we ﬁrst compute the character table. gap> irr:=Irr(G); [ Character( CharacterTable( Q8 ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Q8 ), [ 1, 1, -1, 1, -1 ] ), Character( CharacterTable( Q8 ), [ 1, -1, 1, 1, -1 ] ), Character( CharacterTable( Q8 ), [ 1, -1, -1, 1, 1 ] ), Character( CharacterTable( Q8 ), [ 2, 0, 0, -2, 0 ] )

2.6. More examples 50 Chapter 2. Primitive central idempotents of rational group algebras

So Q8 has the following character table. {1}{xy, x3y}{x, x3}{x2}{y, x2y}

χ1 1 1 1 1 1 χ2 1 1 −1 1 −1 χ3 1 −1 1 1 −1 χ4 1 −1 −1 1 1 χ5 2 0 0 −2 0 Now we can compute the kernel K of the characters and search for a H such that (H,K) is a strong Shoda pair. Remember that by Proposition 2.36, it is enough to ﬁnd one such H. gap> List(irr,KernelOfCharacter); [ Group([ x, y ]), Group([ x ]), Group([ x^-2, y ]), Group([ x^-2, y*x^-1 ]), Group([ ]) ] gap> l:=List(last,x->SearchingKForSSP(QG,x)); [ [ [ Group([ x, y ]), Group([ x, y ]) ], (1/8)*+(1/8)*x+(1/8)*y+(1/8)*x^2+(1/8)*x*y+(1/8)*x^ 3+(1/8)*x^2*y+(1/8)*x^3*y ], [ [ Group([ x, y ]), Group([ x ]) ], (1/8)*y^2*x^2+(1/8)*y^2*x^3+(-1/ 8)*y+(1/8)*y^2+(-1/8)*y*x^3+(1/8)*y^2*x+(-1/8)*y*x^2+(-1/8)*y*x ] , [ [ Group([ x, y ]), Group([ x^-2, y ]) ], (1/8)*+(-1/ 8)*x+(1/8)*y+(1/8)*x^2+(-1/8)*x*y+(-1/8)*x^-1+(1/8)*x^2*y+(-1/ 8)*x^-1*y ], [ [ Group([ x, y ]), Group([ x^-2, y*x^-1 ]) ], (1/8)*y^2*x^-2+(-1/ 8)*y^2*x^-1+(-1/8)*y+(1/8)*y^2+(1/8)*y^2*x^-2*y*x^-1+(-1/8)*y*x^ -2*y*x^-1+(-1/8)*y*x^-2+(1/8)*y^3*x^-1 ], [ [ Group([ x ]), Group([ ]) ], (1/2)*x^4+(-1/2)*x^2 ] ] gap> rc.("PrimitiveCentralIdempotents")[1]=l[1][2]; true gap> rc.("PrimitiveCentralIdempotents")[2]=l[4][2]; true gap> rc.("PrimitiveCentralIdempotents")[3]=l[3][2]; true gap> rc.("PrimitiveCentralIdempotents")[4]=l[2][2]; true gap> rc.("PrimitiveCentralIdempotents")[5]=l[5][2]; true

2 Using the possibilities of GAP, one discover that (Q8,Q8) = eQ(χ1), (Q8, hx , xyi) = eQ(χ4), 2 (Q8, hx , yi) = eQ(χ3), (Q8, hxi) = eQ(χ2) and (hxi, 1) = eQ(χ5).

Remember that the quaternion group Q8 and the dihedral group D8 have the same complex character table. Therefore, the complex group algebras CQ8 and CD8 are isomorphic. We show that this is not the case for the rational group algebras QQ8 and QD8.

4 2 −1 Example 2.39. Consider the dihedral group of order 8, D8 = ha, b | a = b = 1, bab = a i. Since this group is supersolvable, we can compute the Wedderburn decomposition of QD8 using the strong Shoda pairs of D8. We do this in GAP. gap> F:=FreeGroup("a","b");;a:=F.1;;b:=F.2;; gap> G:=F/[a^4,b^2,b*a*b*a];;

2.6. More examples 51 Chapter 2. Primitive central idempotents of rational group algebras gap> QG:=GroupRing(Rationals,G);; gap> SimpleFactorsFromSSP(G); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 2, Rationals ] ]

We see that QD8 ' Q ⊕ Q ⊕ Q ⊕ Q ⊕ M2(Q), and hence QD8 is not isomorphic to QQ8. Now we give an example of a cyclic group.

Example 2.40. Consider G = hxi, the cyclic group of order 6. We are interested in the primitive central idempotents of QG. gap> F:=FreeGroup("x");; gap> x:=F.1;; gap> G:=F/[x^6];; gap> QG:=GroupRing(Rationals,G);; gap> rc:=StrongShodaPairsAndIdempotents(QG); rec ( StrongShodaPairs := [ [ , ], [ , Group([ x^2 ]) ], [ , Group([ x^3 ]) ], [ , Group([ x^6 ]) ] ], PrimitiveCentralIdempotents := [ (1/6)*+(1/6)*x^3+(1/6)*x^2+(1/6)*x^5+(1/6)*x^4+(1/6)*x, (1/6)*+(-1/6)*x^3+(1/6)*x^2+(-1/6)*x^-1+(1/6)*x^-2+(-1/ 6)*x, (1/3)*x^6+(1/3)*x^3+(-1/6)*x^2+(-1/6)*x^5+(-1/6)*x^4+(-1/6)*x , (1/3)*x^-12+(-1/3)*x^-9+(-1/6)*x^-10+(1/6)*x^-7+(-1/6)*x^-8+(1/ 6)* x^-5 ] )

1 2 3 4 5 We ﬁnd the primitive central idempotents e1 = (G, G) = 6 (1 + x + x + x + x + x ), 2 1 2 3 4 5 3 1 2 3 4 5 e2 = (G, hx i) = 6 (1 − x + x − x + x − x ), e3 = (G, hx i) = 6 (2 − x − x + 2x − x − x ) 1 2 3 4 5 and e4 = (G, 1) = 6 (2 + x − x − 2x − x + x ). From these idempotents, we can compute the Wedderburn decomposition. gap> SimpleFactorsFromSSP(G); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(3) ] ]

We see that

QG = QGe1 ⊕ QGe2 ⊕ QGe3 ⊕ QGe4 = 2Q ⊕ 2Q(ξ3) We will give some more examples of abelian-by-supersolvable groups G and we will compute the Wedderburn decompositions of QG. Examples 2.41 and 2.42 can be found in [OAdR03].

2.6. More examples 52 Chapter 2. Primitive central idempotents of rational group algebras

Example 2.41. There are only six non-isomorphic indecomposable groups of order 54, namely the groups

27 2 b −1 G1 = ha, b | a = b = 1, a = a i, 9 6 b 5 G2 = ha, b | a = b = 1, a = a i, 9 3 2 b −1 G3 = ha1, a2, b | a1 = a2 = b = 1, [a1, a2] = 1, ai = ai i, 3 6 b −1 b −1 G4 = ha1, a2, b | ai = b = 1, [a1, a2] = 1, a1 = a1 , a2 = a1a2 i, 3 2 b −1 G5 = ha1, a2, a3, b | ai = b = 1, [ai, aj] = 1, ai = ai i, 3 2 a3 b −1 b −1 G6 = ha1, a2, a3, b | ai = b = 1, [a1, ai] = [a1, b] = 1, a2 = a1a2, a2 = a2 , a3 = a3 i. Using the GAP commands for computing the simple factors from all strong Shoda pairs, we get the following output. Since these groups are all abelian-by-supersolvable, we obtain the full Wedderburn decomposition. G1 [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, NF(9,[ 1, 8 ]), 9, [ [ 2, 8, 0 ] ], [ ] ], [ 1, NF(27,[ 1, 26 ]), 27, [ [ 2, 26, 0 ] ], [ ] ] ]

G2 [ [ 1, Rationals ], [ 1, Rationals ], [ 1, CF(3) ], [ 2, CF(3) ], [ 1, CF(3) ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 3, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ] ]

G3 [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, NF(9,[ 1, 8 ]), 9, [ [ 2, 8, 0 ] ], [ ] ], [ 1, NF(9,[ 1, 8 ]), 9, [ [ 2, 8, 0 ] ], [ ] ], [ 1, NF(9,[ 1, 8 ]), 9, [ [ 2, 8, 0 ] ], [ ] ] ]

G4 [ [ 1, Rationals ], [ 1, Rationals ], [ 1, CF(3) ], [ 2, CF(3) ], [ 1, CF(3) ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 3, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ] ]

G5 [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ],

2.6. More examples 53 Chapter 2. Primitive central idempotents of rational group algebras

[ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ] ]

G6 [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 3, CF(3) ], [ 3, CF(3) ] ] Notice that the tuples in the lists are of one of the following two forms:

[n, K][n, K, k, [[2, k − 1, 0]], []] for n and k integers and K a ﬁeld. Using Proposition 2.37, we can describe the simple factors. In the ﬁrst case the list corresponds to the algebra Mn(Q(ξk)) = Mn(K). In the other case, it corresponds to Mn(Ak) where

−1 2 Ak = Q(ξk)(g | ξkg = gξk , g = 1).

−1 2 Notice that Ak is a quaternion algebra over its center K = Q(ξk + ξk ). Since g = 1, g + 1 is a zero divisor of Ak and therefore Ak is not a division ring. Using Proposition 1.37, we conclude that Ak ' M2(K) and Mn(Ak) = M2n(K). Notice that for k a divisor of 4 or 6, −1 ξk + ξk ∈ Q and hence the center of Ak is Q. Collecting this information, we obtain the Wedderburn decomposition of the respective rational group rings:

−1 −1 QG1 ' 2Q ⊕ M2(Q) ⊕ M2(Q(ξ9 + ξ9 )) ⊕ M2(Q(ξ27 + ξ27 )), QG2 ' 2Q ⊕ 2Q(ξ3) ⊕ M2(Q) ⊕ M2(Q(ξ3)) ⊕ M6(Q), −1 QG3 ' 2Q ⊕ 4M2(Q) ⊕ 3M2(Q(ξ9 + ξ9 )), QG4 ' 2Q ⊕ 2Q(ξ3) ⊕ M2(Q) ⊕ M2(Q(ξ3)) ⊕ M6(Q), QG5 ' 2Q ⊕ 13M2(Q), QG6 ' 2Q ⊕ 4M2(Q) ⊕ 2M3(Q(ξ3)).

Notice that QG2 ' QG4 and that the previous algebras all have trivial twistings. We will now see an example with a simple factor with non-trivial twisting.

Example 2.42. Consider the following abelian-by-supersolvable group of order 48:

G = ha, b, c | a12 = b2a6 = c2a6 = 1, ab = a−1, ac = a7, bc = ba9i.

We use GAP to compute the Wedderburn decomposition of QG. gap> F:=FreeGroup("a","b","c");; a:=F.1;; b:=F.2;; c:=F.3;; gap> G:=F/[a^12,b^2*a^6,c^2*a^6,b^(-1)*a*b*a,c^(-1)*a*c*a^(-7), > c^(-1)*b*c*a^(-9)*b];; gap> SimpleFactorsFromSSP(G); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 2, Rationals ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ], [ 2, CF(3) ], [ 1, Rationals, 6, [ [ 2, 5, 0 ] ], [ ] ], [ 1, NF(8,[ 1, 7 ]), 8, [ [ 2, 7, 4 ] ], [ ] ], [ 1, Rationals, 12, [ [ 2, 11, 6 ], [ 2, 7, 6 ] ], [ [ 3 ] ] ] ]

2.6. More examples 54 Chapter 2. Primitive central idempotents of rational group algebras

We conclude QG = 4Q ⊕ 3M2(Q) ⊕ M2(Q(ξ3)) ⊕ A ⊕ B, where

−1 2 4 A = Q(ξ8)(g | ξ8g = gξ8 , g = ξ8 = −1), −1 2 2 B = Q(ξ12)(g, h | ξ12g = gξ12 , ξ12h = −hξ12, g = h = −1, hg = ghξ4). √ −1 Notice that the center of A is Q(ξ8 + ξ8 ) = Q( 2) and A has the following presentation as algebra over its center: ! √ √ 2 2 −1, −1 A = Q( 2)(g, i = ξ4 | i = g = −1, ig = −gi) = √ = H(Q( 2)). Q( 2) √ Since the Hamiltonian quaternion algebra over Q( 2) is a division ring, this is of the desired form. However it is more diﬃcult to describe B as a matrix ring over a division ring. This needs more sophisticated methods and this example shows the limitations of the method.

Now we will provide some examples on primitive central idempotents of rational group rings. Let G be a ﬁnite group and let E be the set of primitive central idempotents of QG. Set

E1 = {αe(G, H, K) ∈ E | (H,K) is a Shoda pair of G};

E2 = {e(G, H, K) ∈ E | (H,K) is a Shoda pair of G};

E3 = {e(G, H, K) ∈ E | (H,K) is a strong Shoda pair of G};

E4 = {e(G, H, K) ∈ E | (H,K) satisﬁes conditions 1 and 2 of Corollary2 .19}.

Then E4 ⊆ E3 ⊆ E2 ⊆ E1 ⊆ E. Furthermore E = E1 if and only if G is monomial, E = E3 if and only if G is abelian-by-supersolvable and E = E4 if G is metabelian. We will see some examples where E3 6= E4, E2 6= E3 and E1 6= E2. Examples 2.43, 2.44 and 2.45 can be (partially) found in [OdRS04].

Example 2.43 (E3 6= E4). All groups of order at most 23 or metabelian. There are two non- isomorphic non metabelian groups of order 24, namely S4 and SL(2, 3). Only S4 is monomial, moreover S4 is abelian-by-supersolvable and therefore E = E1 = E2 = E3. We compute the primitive central idempotents of QS4. gap> QG:=GroupRing(Rationals,SymmetricGroup(4));; gap> pci:=PrimitiveCentralIdempotentsByStrongSP(QG);; gap> Size(pci); 5

There are 5 primitive central idempotents, however a calculation shows that there are only 3 pairs (H,K) of subgroups of S4 satisfying conditions 1 and 2 of Corollary 2.19, namely (S4,S4), (S4,A4) and (A4, h(14)(23), (13)(24)i). So E4 contains the elements (S4,S4), (S4,A4) and e(S4,A4, h(14)(23), (13)(24)i). We compute the strong Shoda pairs of S4. gap> StrongShodaPairs(SymmetricGroup(4)); [ [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) ], [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]) ],

2.6. More examples 55 Chapter 2. Primitive central idempotents of rational group algebras

[ Group([ (1,2)(3,4), (1,3,2,4), (3,4) ]), Group([ (1,2)(3,4), (1,3,2,4) ]) ], [ Group([ (1,2)(3,4), (3,4), (1,3,2,4) ]), Group([ (1,2)(3,4), (3,4) ])], [ Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]) ] ]

One can identify the previous pairs with the ﬁrst, second and last strong Shoda pair. Notice 2 that (1324) = (12)(34). Therefore, the other two primitive central idempotents of QS4 are e(S4,H, h(1324)i) and e(S4,H, h(12), (34)i) where H = h(1324), (34)i. Since S4 is abelian-by-supersolvable, one can also compute the Wedderburn decomposition. gap> SimpleFactorsFromSSP(SymmetricGroup(4)); [ [ 1, Rationals ], [ 1, Rationals ], [ 3, Rationals ], [ 3, Rationals ], [ 1, Rationals, 3, [ [ 2, 2, 0 ] ], [ ] ] ]

Notice that the tuples in the list are of one of the following two forms:

[n, Q] [1, Q, 3, [[2, 2, 0]], []] for n an integer. By Proposition 2.37 the ﬁrst case corresponds to the algebra Mn(Q). In the other case, it corresponds to

−1 2 A = Q(ξ3)(g | ξ3g = gξ3 , g = 1).

−1 Notice that A is a quaternion algebra over its center and this center is Q = Q(ξ3 + ξ3 ), more speciﬁc −1 2 2 1, −3 A = Q(g, i = ξ3 − ξ3 | g = 1, i = −3, ig = −gi) = . Q Since g2 = 1, g + 1 is a zero divisor of A and therefore A is not a division ring. Using Proposition 1.37, we conclude that A ' M2(Q). Collecting this information, we obtain the Wedderburn decomposition of QS4:

QS4 ' 2Q ⊕ M2(Q) ⊕ 2M3(Q).

Example 2.44 (E2 6= E3). Let G = (hx, yi × hbi) o hai where hx, yi ' Q8 is the quaternion group of order 8, hbi ' C7 and hai ' C3. And the action of hai on hx, yi × hbi is given by xa = y, ya = xy and ba = b2. We compute the Shoda pairs and the strong Shoda pairs of G. gap> F:=FreeGroup("x","y","b","a");; x:=F.1;; y:=F.2;; b:=F.3;; a:=F.4;; gap> G:=F/[x^4,x^2*y^2,y^(-1)*x*y*x,b^7,a^3,x^(-1)*b^(-1)*x*b, > y^(-1)*b^(-1)*y*b,y*a*x^(-1)*a^(-1),x*y*a*y^(-1)*a^(-1), > b^2*a*b^(-1)*a^(-1) ];; gap> IsMonomial(G); false gap> QG:=GroupRing(Rationals, G);; gap> rc:=ShodaPairsAndIdempotents(QG);; gap> rc.("ShodaPairs"); [ [ (C7 x Q8) : C3, (C7 x Q8) : C3 ], [ Group([ x, y, b, a ]), C7 x Q8 ], [ Group([ x, y, b ]), Group([ x, b ]) ], [ Group([ x, y, b ]), Group([ x, y ]) ], [ Group([ x, y, b ]), Group([ x ]) ], [ Group([ , y^2, x*b ]), Group([ ]) ] ] gap> StrongShodaPairs(G);

2.6. More examples 56 Chapter 2. Primitive central idempotents of rational group algebras

[ [ (C7 x Q8) : C3, (C7 x Q8) : C3 ], [ (C7 x Q8) : C3, C7 x Q8 ], [ Group([ x, y, b ]), Group([ x, b ]) ], [ Group([ x, y, b ]), Group([ x, y ]) ], [ Group([ x, y, b ]), Group([ x ]) ] ]

Note that G is not monomial and therefore E 6= E1. We see there is one Shoda pair which is not 2 a strong Shoda pair. Note that h1, y , xbi = hxbi. We discover that E1 = E3∪{˙ e(G, hxbi, 1)}, a disjoint union. Now we search for α such that αe(G, hxbi, 1) is an idempotent. gap> H:=rc.("ShodaPairs")[6][1]; Group([ , y^2, x*b ]) gap> K:=rc.("ShodaPairs")[6][2]; Group ([ ]) gap> e:=CentralElementBySubgroups(QG,H,K);; gap > e*e=e; true

We ﬁnd that e(G, hxbi, 1) is already an idempotent and therefore e(G, hxbi, 1) ∈ E2. This means that the G-conjugates of (hxbi, 1) are orthogonal. By Proposition 2.16 it follows that hxbi is not normal in NG(1) = G.

Example 2.45 (E1 6= E2). Let G = (D × C7) o C3 where C7 = hbi and C3 = hai and D = hx, y, z | z4 = 1, x2 = y2 = z2, [x, z] = [y, z] = 1, xy = x−1i.

a a a a 2 And the action of hai on D × C7 is given by x = y, y = xy, z = z and b = b . gap> F:=FreeGroup("x","y","z","b","a");; gap> x:=F.1;;y:=F.2;;z:=F.3;;b:=F.4;;a:=F.5;; gap> G:=F/[z^4,x^2*y^(-2),x^2*z^2,x^(-1)*z^(-1)*x*z,y^(-1)*z^(-1)*y*z, > y^(-1)*x*y*x,b^7,a^3,x^(-1)*b^(-1)*x*b,y^(-1)*b^(-1)*y*b, > z^(-1)*b^(-1)*z*b,a^(-1)*x*a*y^(-1),a^(-1)*y*a*y^(-1)*x^(-1), > a^(-1)*z*a*z^(-1),a^(-1)*b*a*b^(-2)];; gap> IsMonomial(G); false gap> QG:=GroupRing(Rationals,G);; gap> StrongShodaPairs(G); [ [ ((C7 x Q8) : C3) : C2, ((C7 x Q8) : C3) : C2 ], [ ((C7 x Q8) : C3) : C2, (C7 x Q8) : C3 ], [ ((C7 x Q8) : C3) : C2, Group([ x, y, z, b ]) ], [ Group([ x, y, z, b ]), Group([ x^-2, y*x^-1, z, b ]) ], [ Group([ x, y, z, b ]), Group([ x^-2, y*x^-1, z*x^-1, b ]) ], [ ((C7 x Q8) : C3) : C2, Group([ x, y, b ]) ], [ Group([ x, y, z, b ]), Group([ x, y, z ]) ], [ Group([ x, y, z, b ]), Group([ x^-2, y*x^-1, z ]) ], [ Group([ x, y, z, b ]), Group([ x^-2, y*x^-1, z*x^-1 ]) ], [ Group([ x, y, z, b ]), Group([ x, y ]) ] ] gap> Size(last); 10 gap> rc:=ShodaPairsAndIdempotents(QG);; gap> rc.("ShodaPairs"); [ [ ((C7 x Q8) : C3) : C2, ((C7 x Q8) : C3) : C2 ], [ Group([ x, y, z, b, a ]), (C7 x Q8) : C3 ], [ Group([ x, y, z, b, a ]), Group([ x, y, z, b ]) ], [ Group([ x, y, z, b ]), Group([ x^-2, y*x^-1, z, b ]) ],

2.6. More examples 57 Chapter 2. Primitive central idempotents of rational group algebras

[ Group([ x, y, z, b ]), Group([ x^-2, y*x^-1, z*x^-1, b ]) ], [ Group([ x, y, z, b, a ]), Group([ x, y, b ]) ], [ Group([ x, y, z, b ]), Group([ x, y, z ]) ], [ Group([ x, y, z, b ]), Group([ x^-2, y*x^-1, z ]) ], [ Group([ x, y, z, b ]), Group([ x^-2, y*x^-1, z*x^-1 ]) ], [ Group([ x, y, z, b ]), Group([ x, y ]) ], [ Group([ x^-2, y*x^-1, z, b ]), Group([ x*y*z ]) ] ] gap> Size(last); 11 gap> H:=rc.("ShodaPairs")[11][1];; K:=rc.("ShodaPairs")[11][2];; gap> e:=CentralElementBySubgroups(QG,H,K);; gap > e*e=e; false gap> 1/3*e*1/3*e=1/3*e; true gap> IsInE2(QG,rc.("PCIsBySP")[11]); false

We discover that E1 has 11 elements and E3 has 10 elements. Therefore E2 = E1 or E2 = E3. 1 However the element in E1 \E3 is of the form 3 e(G, H, K), with H = hxy, z, bi and K = hxyzi, and can not be rewritten as an element in E2. Therefore E1 6= E2 = E3.

By Theorem 2.25 if G is abelian-by-supersolvable then E = E3 and therefore for every primitive central idempotent e ∈ QG there is a linear character χ of a subgroup H of G G with kernel K such that e = eQ(χ ) = e(G, H, K), by Theorem 2.11, that is [Cen((H,K)) : H] = [Q(χ): Q(χG)]. Since abelian-by-supersolvable groups are monomial, if the equality [Cen((H,K)) : H] = [Q(χ): Q(χG)] would hold for every Shoda pair (H,K) and every character χ of H with kernel K then this fact would follow without Theorem 2.25. The next example shows that this is not the case. This example can be found in [OdRS04].

Example 2.46. Let C3 = hai,C2 = hbi,Q8 = hx, yi and consider G = C3 o (Q8 o C2) where the actions are given by

ab = a, xb = x−1, yb = yx, ax = a, ay = a−1.

Consider K = hbi and H = ha, b, x2i. gap> F:=FreeGroup("a","b","x","y");; a:=F.1;;b:=F.2;;x:=F.3;;y:=F.4;; gap> G:=F/[a^3,b^2,x^4,x^2*y^2,y^(-1)*x*y*x,b^(-1)*x*b*x, > b^(-1)*y*b*x^(-1)*y^(-1),x^(-1)*a*x*a^(-1),y^(-1)*a*y*a, > a^(-1)*b^(-1)*a*b];; gap> IsSupersolvable(G); true gap> rc:=ShodaPairsAndIdempotents(QG);; gap> rc.("ShodaPairs"); [ [ , ], [ Group([ a, b, x, y ]), Group([ a, x, y*b^-1 ]) ], [ Group([ a, b, x, y ]), Group([ a, x, y ]) ], [ Group([ a, b, x, y ]), Group([ a, b, x ]) ], [ Group([ a, x, y ]), Group([ a, x^-2, y ]) ], [ Group([ a, b, x ]), Group([ b, x ]) ], [ Group([ a, b, x ]), Group([ b, x^-2 ]) ], [ Group([ a, b, x ]), Group([ x ]) ],

2.6. More examples 58 Chapter 2. Primitive central idempotents of rational group algebras

[ Group([ a, x, y*b^-1 ]), Group([ a ]) ], [ Group([ a, b, x^-2 ]), Group([ b ]) ] ] gap> List(rc.("ShodaPairs"),x->IsStrongShodaPair(G,x[1],x[2])); [ true, true, true, true, true, true, true, true, true, false ] gap> H:=rc.("ShodaPairs")[10][1]; Group([ a, b, x^-2 ]) gap> K:=rc.("ShodaPairs")[10][2]; Group ([ b ]) gap> Normalizer(G,K); Group([ a, b, x^-2 ]) We see that (H,K) is a Shoda pair for H = ha, b, x2i and K = hbi and therefore the linear 2 character of H, χ : H → C given by χ(ax ) = ξ6 and χ(b) = 1 induces an irreducible monomial G character χ of G. We noticed that H = NG(K) and that (H,K) is not a strong Shoda pair. Therefore, by Proposition 2.16, the G-conjugates of (H,K) should not be orthogonal. We can verify this in GAP. gap> e:=CentralElementBySubgroups(QG,H,K); (2/3)*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(-2/3)*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(-1/3)*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(1/ 3)*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(-1/3)*x^-2*a^2*x^-2*a^ 2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^ 2+(1/3)*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2 gap > e*e; (4/3)*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(-4/3)*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(-2/3)*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(2/3)*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(-2/3)*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2+(2/3)*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^ -2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2*x^-2*a^2 gap> e*e=2*e; true

e(G,H,K) G e(G,H,K) Indeed, we see that e(G, H, K) is not idempotent, but 2 is. Therefore eQ(χ ) = 2 . G However we noticed that G is supersolvable and therefore, by Theorem 2.25, eQ(χ ) = e(G, H1,K1) for some strong Shoda pair (H1,K1) of G. We can ﬁnd this pair using GAP. gap> StrongShodaPairs(G); [ [ , ], [ , Group([ a, x, y*b^-1 ]) ], [ , Group([ a, x, y ]) ], [ , Group([ a, b, x ]) ], [ Group([ a, x, y ]), Group([ a, x^-2, y ]) ] , [ Group([ a, b, x ]), Group([ b, x ]) ], [ Group([ a, b, x ]), Group([ b, x^-2 ]) ],

2.6. More examples 59 Chapter 2. Primitive central idempotents of rational group algebras

[ Group([ a, b, x ]), Group([ x ]) ], [ Group([ a, x, y*b^-1 ]), Group([ a ]) ], [ Group([ a, x, y*a*y^-1, y*b*y^-1*b^-1, y*x*y^-1, y^2, b*a*b^-1, b^2, b*x*b^-1, b*y*a*y^-1*b^-1, b*y*b*y^-1, b*y*x*y^-1*b^-1, b*y^2*b^-1 ]), Group([ ]) ] ] gap> AsList(last[10][1]); [ , x, x^2, a, x^3, a*x, a*x^2, a^2, a*x^3, a^2*x, a^2*x^2, a^2*x^3 ]

Notice that the only pair that did not appear in the list of Shoda pairs is the last one. So this must be the wanted pair (H1,K1). The pair can be simpliﬁed and we get H1 = ha, xi and K1 = {1}. Now we compute the primitive central idempotents of the smallest non abelian-by-supersolvable group (which coincides with the smallest non monomial group) in terms of the elements of the form e(G, H, K). This example can be (partially) found in [OdRS04].

Example 2.47. The minimal non monomial (non abelian-by-supersolvable) group is SL(2, 3) a a which is isomorphic to G = Q8 o C3 where Q8 = hx, yi,C3 = hai and x = y and y = xy. gap> F:=FreeGroup("x","y","a");; x:=F.1;;y:=F.2;;a:=F.3;; gap> G:=F/[x^4,x^2*y^2,y^(-1)*x*y*x,a^3,a^(-1)*x*a*y^(-1), > a^(-1)*y*a*y^(-1)*x^(-1)];; gap> x:=G.1;;y:=G.2;;a:=G.3;; gap> QG:=GroupRing(Rationals,G);; gap> pci:=PrimitiveCentralIdempotentsByCharacterTable(QG);; gap> Size(pci); 5 gap> IsSolvable(G); true gap> StructureDescription(FactorGroup(G,Subgroup(G,[x^2]))); "A4" gap> IsMetabelian(AlternatingGroup(4)); true

Using GAP, we discover that G has 5 primitive central idempotents. Notice that G/hx2i = hx, yi o hai ' A4, which is metabelian and therefore the primitive central idempotents of QA4 are (A4,A4), (A4, hx, yi) and e(A4, hx, yi, hxi). Thus, for G, E4 has at least 3 elements: Ü Ý Ü Þ2 Ý f1 = (G, G) = G, f2 = (G, Q8) = Q8 − G and f3 = e(G, Q8, hxi) = hx i − Q8. Let Þ2 2 e = 1 − f1 − f2 − f3 = 1 − hx i = (hx i, 1). However e is not a primitive central idempotent, namely e is the sum of the following primitive central idempotents of QG e(G, B, A) (1 − x2)(4 + (1 + (x + y + xy))a + (1 − (x + y + xy))a2) e = = , 1 2 12 e(G, B, 1) − e(G, B, A) (1 − x2)(2 − (1 + (x + y + xy))a − (1 − (x + y + xy))a2) e = = , 2 4 12 where A = hai and B = hx2ai. We can verify that these are the primitive central idempotents of QG. gap> Q8:=Subgroup(G,[x,y]);; gap> f1:=IdempotentBySubgroups(QG,G,G);; gap> f2:=IdempotentBySubgroups(QG,G,Q8);;

2.6. More examples 60 Chapter 2. Primitive central idempotents of rational group algebras gap> f3:=CentralElementBySubgroups(QG,Q8,Subgroup(G,[x]));; gap> A:=Subgroup(G,[a]);; gap> B:=Subgroup(G,[x^2*a]);; gap> e1:=CentralElementBySubgroups(QG,B,A)/2;; gap> e2:=1/4*(CentralElementBySubgroups(QG,B,Subgroup(G,[]))- > CentralElementBySubgroups(QG,B,A));; gap> pci[1]=f1; true gap> pci[2]=f2; true gap> pci[3]=e2; true gap> pci[4]=e1; true gap> pci[5]=f3; true

Since G is not monomial, E 6= E1 and therefore either e1 or e2 do not belong to E1. Indeed, there are only 3 elements in E1. gap> ShodaPairsAndIdempotents(QG).("ShodaPairs"); [ [ SL(2,3), SL(2,3) ], [ SL(2,3), Group([ x, y ]) ], [ Group([ x, y ]), Group([ x^-2, y ]) ] ]

We will prove that e1 and e2 do not belong to E1. Assume that ei = αe(G, K, H) for i = 1 or 2, α ∈ Q and some Shoda pair (H,K) of G. Then α = [CenG((H,K)):H] for some linear character χ of H with kernel K and hence [Q(χ):Q(χG)] G G |H|[Q(χ): Q(χ )]ei ∈ ZG. Thus |H|[Q(χ): Q(χ )] is a multiple of 12. Let h = |H|. Then [ (χ): (χG)] = [Q(χ):Q] = [Q(ξ[H:K]):Q] divides φ([H : K]) and φ([H : K]) divides φ(h). Q Q [Q(χG):Q] [Q(χG):Q] Therefore hφ(h) is a multiple of 12. We deduce that h = 6, 12 or 24. If h 6= 6, then H is a normal subgroup of G. Since the normal subgroups of G are 1, hx2i, hx, yi and G, and none of them are of order 12, we have that H = G and so K ¡ G. This implies that (G, K) is a strong Shoda pair and hence α = 1. However ei 6= e(G, G, K) for any normal subgroup K of G. Thus h = 6. Conjugating by an appropriate element of G one may assume without loss of generality that H contains A. The only subgroup of G of order 6 that contains A is B, so that H = B. However (B,K) is not a Shoda pair for any subgroup K of B because [B, x] ∩ B = {1} and x∈ / B.

Example 2.47 shows that conditions 2(a)-2(c) and 3(A)-3(C) of Corollary 2.26 are not enough to ensure that e(G, H, K) is a primitive central idempotent, even if G is solvable. Indeed, however (hx2i, 1) does satisfy the conditions 2(a)-2(c) and 3(A)-3(C), e = (hx2i, 1) is not a primitive central idempotent of QG.

2.7 The primitive central idempotent associated to an irreducible character

In this section we will give a description of every primitive central idempotent eQ(χ) of a rational group algebra QG corresponding to an irreducible character χ of a ﬁnite group G as a rational linear combination of elements of the form e(G, Hi,Ki), with each (Hi,Ki) a strong

2.7. The primitive central idempotent associated to an irreducible character 61 Chapter 2. Primitive central idempotents of rational group algebras

Shoda pair in a subgroup of G, or equivalently, Ki is a normal subgroup of Hi with Hi/Ki cyclic. To do this we will use Brauer’s Theorem on Induced Characters. This section is based on a recent article of Jespers, Olteanu and del R´ıo[JOdR10]. The following Theorem is Brauer’s Theorem on Induced Characters [BT55].

Theorem 2.48 (Brauer)P . Every complex character χ of a finite group G is a Z-linear G G combination χ = i aiθi , ai ∈ Z, of characters θi induced from linear characters θi of elementary subgroups Mi of G, where by an elementary subgroup of G we mean one which is a direct product of a cyclic group and a p-group for some prime p. We will use the following definition of field trace.

Definition 2.49. Let L/K be a finite Galois extension. The field trace of this extension is defined as X trL/K (α) = σ(α), σ∈Gal(L/K) for α ∈ L. Note that this map is K-linear. Let G be a group. Then we can extend this trace to the group ring LG by the rule X X trL/K αgg = trL/K (αg)g. g∈G g∈G

This extension remains K-linear.

Theorem 2.50. Let G be a ﬁnite group of order n and χ an irreducible character of G.

Then the primitive central idempotent eQ(χ) of QG associated to χ is of the form X χ(1) [Q(ξn): Q(ψi)] eQ(χ) = ai e(G, Hi,Ki), [Q(ξn): Q(χ)] i [G : CenG((Hi,Ki))] where ai ∈ Z, (Hi,Ki) are strong Shoda pairs of subgroups of G (equivalently Ki is a normal subgroup of Hi with Hi/Ki cyclic) and ψi are linear characters of Hi with kernel Ki. Proof. For every χ ∈ Irr(G), by (2.1) we have that X eQ(χ) = σ(e(χ)) = trQ(χ)/Q(e(χ)). σ∈Gal(Q(χ)/Q)

The interpretation of eQ(χ) as a trace, suggests the following useful notation for the next arguments. For any ﬁnite Galois extension F of Q containing Q(χ), let X F eQ(χ) = σ(e(χ)) = trF/Q(e(χ)). σ∈Gal(F/Q)

Hence e (χ) = eQ(χ)(χ) = 1 eF (χ) for every ﬁnite Galois extension F of containing Q Q [F :Q(χ)] Q Q Q(χ), since Gal(Q(χ)/Q) ' Gal(F/Q)/Gal(F/Q(χ)) by the Fundamental TheoremP of Galois G Theory. Using Brauer’s Theorem on Induced Characters, we can write χ = i aiψi , with ψi

2.7. The primitive central idempotent associated to an irreducible character 62 Chapter 2. Primitive central idempotents of rational group algebras linear characters of elementary subgroups Hi with kernel Ki and ai ∈ Z. By Theorem 1.28 Q(χ) ⊂ Q(ξn) and Q(ψi) ⊂ Q(ξn) and X χ(1) eQ(ξn)(χ) = a eQ(ξn)(ψG), Q i Q i i [G : Hi] P P P G χ(1) G χ(1) G since trQ(ξn)/Q is Q-linear and e( i aiψi ) = i ai G e(ψi ) = i ai e(ψi ). We will ψi (1) [G:Hi] compute eQ(ξn)(ψG) as in the proof of Theorem 2.11. Q i Put ei = e(ψi). We know that Aut(C) acts on the left and G acts on the right on ψi and on ei (by composition and by conjugation respectively) and that their actions are compatible. As in the proof of Theorem 2.11, we can consider Aut(C) × G acting on the left on the set of −1 irreducible characters of subgroups of G (and similarly on the ei’s) by (σ, g) · ψi = σ · ψi · g −1 (and (σ, g) · ei = σ · ei · g ). Let Gal(Q(ξn)/Q) = {σ1, . . . , σl} and Ti = {gi1, . . . , gim} be aP right transversal of Hi in G m G. Similar as in the proof of Theorem 2.11, we have that e(ψi ) = k=1 ei · gik. Now compute

Xl Xl Xm eQ(ξn)(ψG) = σ e(ψG) = σ · e · g Q i j i j i ik j=1 j=1 k=1 Xm

= trQ(ξn)/Q(ei) · gik k=1 Xm

= [Q(ξn): Q(ψi)]trQ(ψi)/Q(ei) · gik. k=1

As in Theorem 2.11, trQ(ψi)/Q(ei) = (Hi,Ki) and hence

Xm eQ(ξn)(ψG) = [ (ξ ): (ψ )](H ,K ) · g Q i Q n Q i i i ik k=1 = [Q(ξn): Q(ψi)][CenG((Hi,Ki)) : Hi]e(G, Hi,Ki). Note that the last equation holds since

Xm gik (Hi,Ki) = [CenG((Hi,Ki)) : Hi]e(G, Hi,Ki) k=1 by similar arguments as in Theorem 2.11. Then

1 X χ(1) e (χ) = a eQ(ξn)(ψG) Q i Q i [Q(ξn): Q(χ)] i [G : Hi] 1 X χ(1) = ai [Q(ξn): Q(ψi)][CenG((Hi,Ki)) : Hi]e(G, Hi,Ki) [Q(ξn): Q(χ)] i [G : Hi] X χ(1) [Q(ξn): Q(ψi)] = ai e(G, Hi,Ki). [Q(ξn): Q(χ)] i [G : CenG((Hi,Ki))]

It remains to prove that the pairs (Hi,Ki) are strong Shoda pairs of subgroups of G. But this is satisﬁed since this condition is clearly equivalent with saying that Ki is a normal subgroup of Hi with Hi/Ki cyclic. This proves the statement. 2

2.7. The primitive central idempotent associated to an irreducible character 63 Chapter 2. Primitive central idempotents of rational group algebras

Notice that the formula from Theorem 2.50 for the computation of the primitive central idempotent eQ(χ) of QG associated to an irreducible character χ of G coincides with the formula from Theorem 2.11 in the case that χ is a monomial irreducible character of G induced from one linear character ψ of a subgroup H with kernel K. To implement the formula from Theorem 2.50 in GAP, on has to consider all strong Shoda pairs in subgroups of G that contribute to the description of a primitive central idempotent of QG. However, on can reduce the search to representatives given by a relation between such pairs of subgroups. Indeed, in [OdRS06] the following Proposition is stated.

Proposition 2.51. Let (H1,K1) and (H2,K2) be two Shoda pairs of a ﬁnite group G and let α1, α2 ∈ Q be such that ei = αie(G, Hi,Ki) is a primitive central idempotent of QG for g g i = 1, 2. Then e1 = e2 if and only if there is a g ∈ G such that H1 ∩ K2 = K1 ∩ H2.

Proof. Reformulating [CR88, Th. 45.6], we get that, for all characters ψi of Hi with kernel G G g Ki, ψ1 = ψ2 if and only if there exist a g ∈ G such that for all h ∈ H1 ∩ H2 it holds that −1 ψ1(ghg ) = ψ2(h). G Now, let χi be a linear character of Hi with kernel Ki, then ei = eQ(χi ), by Theorem 2.11. Recall from the proof of Corollary 2.13 that e1 = e2 if and only if there exists a σ ∈ Aut(C) G G G such that χ1 = σχ2 = (σχ2) . Assume that e1 = e2, then, following the above noted Theorem, there exist a g ∈ G g −1 such that for all h ∈ H1 ∩ H2 it holds that χ1(ghg ) = σ(χ2(h)). Hence the kernels of g −1 0 g ρ : H1 ∩ H2 → C : h 7→ χ1(ghg ) and ρ : H1 ∩ H2 → C : h 7→ σ(χ2(h)) are equal, so g g K1 ∩ H2 = H1 ∩ K2. g g g −1 Assume K1 ∩H2 = H1 ∩K2, which are the kernels of ρ : H1 ∩H2 → C : h 7→ χ1(ghg ) and 00 g g−1 g ρ : H1 ∩ H2 → C : h 7→ χ2(h) respectively. Then χ1(H1 ∩ H2 ) ' χ2(H1 ∩ H2). Moreover −1 g there exists a σ ∈ Aut(C) such that χ1(ghg ) = σ(χ2(h)) for all h ∈ H1 ∩ H2. Again by the G G above noted Theorem, we get that χ1 = (σχ2) and hence e1 = e2. 2

Also, to be able to implement the formula from Theorem 2.50 in GAP, we would like to be able to give a bound for the integers ai used in the formula. To improve the runtime of the algorithm, we would like to give more information on the pairs of groups (Hi,Ki) that one has to consider in the description of eQ(χ). This is not the goal for my thesis, but I would like to do some research on this implementation in the near future. The next example shows that, in general, more than one Shoda pair in a subgroup of G can appear. Therefore we do not have any control on the rational coeﬃcients occurring in the formula.

Example 2.52. Remember that for monomial groups, all primitive central idempotents are realized as elements of the form αe(G, H, K), with α ∈ Q, for some Shoda pair (H,K) of G. However, we have seen that for the smallest non-monomial group SL(2, 3) in Example 2.47, the two primitive central idempotents corresponding to the non-monomial characters 1 1 1 2 are e1 = 2 e(G, B, A) and e2 = 4 e(G, B, 1)− 4 e(G, B, A), with A = hai and B = hx ai. We see that e1 is indeed a rational linear multiple of only one term e(G, H, K) with (H,K) a strong Shoda pair in a subgroup of G, but a computer search shows that e2 cannot be written as a rational linear multiple of some e(G, H, K) with (H,K) a strong Shoda pair in a subgroup of G.

2.7. The primitive central idempotent associated to an irreducible character 64 Chapter 3

Primitive central idempotents of ﬁnite group algebras

Analogously to the rational group algebra, we compute the Wedderburn decomposition and the primitive central idempotents of a semisimple ﬁnite group algebra of an abelian-by- supersolvable group G from certain pairs of subgroups of G. This chapter is based on [BAdR07].

3.1 Some notations and background

In this chapter, F = Fq denotes a finite field with q (a prime power) elements and G a finite group of order n such that FG is semisimple, or equivalently (by Maschke’s theorem) (q, n) = 1. The algebraic closure of F is denoted by F. For every positive integer k coprime with q, ξk denotes a primitive kth root of unity in F and ok denotes the multiplicative order of q modulo k. Since finite fields are classified by size and the non-zero elements in a field of m m size p form a group under multiplication of order p − 1, it is easy to see that F(ξk) ' Fqok , the field of order qok . We assume that all the characters of any finite group are considered as characters in F. Similarly as in Chapter2, we define, for an irreducible character χ of G, eF(χ) to be the only primitive central idempotent e of FG such that χ(e) 6= 0. The automorphism group Aut(F) acts on FG by acting on the coefficients, that is X X σ agg = σ(ag)g, g∈G g∈G for σ ∈ Aut(F) and ag ∈ F. Following [Yam73], we have the formula X eF(χ) = σe(χ). σ∈Gal(F(χ)/F)

∗ m The group Zn of units of the ring Zn acts on G by mg = g . Let Q denote the subgroup ∗ of Zn generated by the class of q and consider Q acting on G by restriction of the previous action. The q-cyclotomic classes of G are the orbits of G under the action of Q on G. Note that if g ∈ G, then the cardinality of the cyclotomic class Cq(g) containing g is the q q2 qo−1 multiplicative order o of q modulo the order of g and Cq(g) = {g, g , g , . . . , g }. The

65 Chapter 3. Primitive central idempotents of finite group algebras q-cyclotomic classes of the cyclic group (Zn, +) are called the q-cyclotomic classes modulo n. Assume now that G = hgi is cyclic of order n. Then the irreducible characters are all linear and deﬁned by the image of a generator of G. Therefore the set G∗ = Irr(G) of irreducible ∗ characters of G is a group with product (χ1χ2)(g) = χ1(g)χ2(g), for χ1, χ2 ∈ G and g ∈ G. ∗ i i ∗ Furthermore, G ' G : g 7→ (χ : G → F : g 7→ ξn) is an isomorphism and in particular G ∗ i is cyclic. The generators of G are the irreducible characters χ : G → F : g 7→ ξn, such that i ∗ ξn generates all n-th roots of unity. Therefore, the generators of G are precisely the faithful representations of G. ∗ If G is cyclic, then let C(G) = Cq(G) denote the set of q-cyclotomic classes of G that contains generators of G∗.

Let N ¡G such that G/N is cyclic of order k and C ∈ C(G/N). If χ ∈ C and tr = trF(ξk)/F denotes the ﬁeld trace of the Galois extension F(ξk)/F, then we set X X −1 −1 −1 f −1 C (G, N) = |G| tr(χ(gN))g = [G : N] N tr(χ(X))gX , g∈G X∈G/N where gX denotes a representative of X ∈ G/N. Let K ¡ H ≤ G such that H/K is cyclic and C ∈ C(H/K). Then eC (G, H, K) denotes the sum of the diﬀerent G-conjugates of C (H,K). Note that if N ¡ G is such that G/N is cyclic, then C (G, N) does not depend on the choice of χ ∈ C. Indeed, if ψ is another element of C then ψ = χqi , for some i and hence tr(ψ(g)) = tr(χ(g)qi ) = tr(χ(g)), because the Frobenius automorphism, x 7→ xq, belongs to Gal(F(ξk)/F). ∗ m Note also that if g is a generator of G then the map φ : Zn → G given by φ(m)(g) = ξn is a group homomorphism and φ induces a one-to-one correspondence between the q-cyclotomic ∗ classes modulo n contained in Zn and C(G).

3.2 The primitive central idempotents

In this section we will see that for certain groups G and certain subgroups H and K, the elements C (H,K) and eC (G, H, K) will appear as the primitive central idempotents of FG, if FG is semisimple. We will also be able to describe the components in the Wedderburn decomposition associated to those idempotents. Since F is ﬁnite, each Wedderburn component of FG, will be a matrix ring over a ﬁeld. So, we do not have problems with non-commutative division rings. We will need the following Lemma. This Lemma can be found in [Rom06].

Lemma 3.1. The Galois group G of Fqo over Fq is cyclic of order o and generated by the Frobenius automorphism f : x 7→ xq. Proof. The fact that f ∈ G is obvious. Moreover, the o automorphisms

1, f, f 2, . . . , f o−1

m qm are distinct elements of G, because if f = 1, then x = x for all x ∈ Fqo and so Fqm ⊇ Fqo , which implies that m ≥ o. Finally, since |G| = o, we see that G = hfi. 2

3.2. The primitive central idempotents 66 Chapter 3. Primitive central idempotents of finite group algebras

The following Proposition shows that the primitive central idempotents of FG, for G abelian, are of the form C (G, N), with N ¡ G such that G/N is cyclic. Note that this Proposition is similar to Proposition 2.4.

Proposition 3.2. If G is a finite abelian group of order n and F is a finite field of order q such that (q, n) = 1, then the map (N,C) 7→ C (G, N) is a bijection from the set of pairs (N,C) with N ¡ G, such that G/N is cyclic, and C ∈ C(G/N) to the set of primitive central idempotents of FG. Further, for every N ¡ G and C ∈ C(G/N), FGC (G, N) ' F(ξk), where k = [G : N]. Proof. If e is a primitive central idempotent of FG, then there is an irreducible character ψ of G such that e = eF(ψ). Since G is abelian, ψ is linear. Let N = ker ψ and let χ be the faithful character of G/N given by χ(gN) = ψ(g). Note that G/N is cyclic as a subgroup of FÒ∗. Since χ is faithful, it is a generator of (G/N)∗ and the cyclotomic class C of (G/N)∗ containing χ belongs to C(G/N) and because of Theorem 1.27 X 1 X X e (ψ) = σe(ψ) = σ(ψ(g))g−1 F |G| σ∈Gal(F(ψ)/F) σ∈Gal(F(ψ)/F) g∈G X 1 −1 = tr(χ(gN))g = C (G, N). |G| g∈G

This shows that the map is surjective and that FGC (G, N) = FGeF(ψ). Since neF(ψ) = eF(ψ) for all n ∈ N and G/N is cyclic of order k, we have that FGeF(ψ) ' F(ξk) = F(ψ). Now, consider C (G, N) with N ¡ G, G/N cyclic and C ∈ C(G/N). Take χ ∈ C and let π : G → G/N be the canonical projection. Then is ψ = χ ◦ π an irreducible character of G, and by the calculations above, we know that eC (G, N) = eF(ψ). Hence, eC (G, N) is a primitive central idempotent of FG and the map is well-deﬁned.

To show that the map is injective, assume that eC1 (G, N1) = eC2 (G, N2) with Ni ¡G, G/Ni cyclic and χi ∈ Ci ∈ C(G/Ni). Let πi : G → G/Ni be the canonical projection and ψi = χi ◦πi.

By the calculation above, one has that eF(ψ1) = eC1 (G, N1) = eC2 (G, N2) = eF(ψ2) and hence F(ψ1) ' FGeF(ψ1) = FGeF(ψ2) ' F(ψ2), hence F(ψ1) = F(ψ2). If K = F(ψi), then there is σ ∈ Gal(K/F) such that ψ2 = σ ◦ ψ1 and hence N1 = ker ψ1 = ker ψ2 = N2. By Lemma 3.1, Gal(K/F) is generated by the Frobenius automorphism and therefore, there is a positive qi qi −1 integer i such that σ(x) = x , for every x ∈ K, and hence ψ2(x) = ψ1(x) . Let now π be −1 −1 qi qi a right inverse of π1 = π2, then χ2 = ψ2 ◦ π = (ψ1 ◦ π ) = χ1 . Thus, C2 = C1. 2

Corollary 3.3. Let G be a ﬁnite group. If N ¡ G is such that G/N is cyclic and C ∈ C(G/N), then C (G, N) is a primitive central idempotent of FG and FGC (G, N) ' F(ξk), where k = [G : N]. Furthermore, if D is another element of C(G/N) then C (G, N) = D(G, N) if and only if C = D. f Proof. The canonical homomorphism G → G/N induces an isomorphism φ : FGN ' f F(G/N). Since C (G, N) ∈ FGN and φ(C (G, N)) = C (G/N, 1) is a primitive central idempotent of F(G/N), also C (G, N) is a primitive central idempotent of FG. Further- more, FGC (G, N) ' F(G/N)C (G/N, 1) ' F(ξk) and C (G, N) = D(G, N) if and only if C (G/N, 1) = D(G/N, 1) if and only if C = D. 2

If H is a subgroup of G and ψ is a linear character of H and g ∈ G, then ψg denotes

3.2. The primitive central idempotents 67 Chapter 3. Primitive central idempotents of finite group algebras the character of Hg given by ψg(hg) = ψ(h). This deﬁnes an action of G on the set of linear characters of subgroups of G. Note that if K = ker ψ, then ker ψg = Kg and therefore the rule ψ 7→ ψg deﬁnes a bijection between the set of linear characters of H with kernel K and the set of linear characters of Hg with kernel Kg. This bijection maps q-cyclotomic classes to q-cyclotomic classes and hence induces a bijection C(H/K) → C(Hg/Kg). The image of C ∈ C(H/K) under this map is denoted as Cg. The following equality is obvious

g g g C (H,K) = Cg (H ,K ) (3.1)

Let K ¡ H ≤ G be such that H/K is cyclic. Then N = NG(H) ∩ NG(K) acts on H/K by conjugation and this induces an action of N on the set of q-cyclotomic classes of H/K. Let i h1K and h2K be generators of H/K and h1K = h2K, then

q qo−1 g qj StabN ({h1K, h1K, . . . h1 K}) = {g ∈ N | h1K = h1 K for some j} i g i qj = {g ∈ N | (h2) K = (h2) K for some j} g i qj i = {g ∈ N | (h2) K = (h2 ) K for some j} g qj ⊇ {g ∈ N | h2K = h2 K for some j} q qo−1 = StabN ({h2K, h2K, . . . h2 K}).

By reversing the roles of h1 and h2, we get that

q qo−1 q qo−1 StabN ({h1K, h1K, . . . h1 K}) = StabN ({h2K, h2K, . . . h2 K}) and hence the stabilizers of all the q-cyclotomic classes of H/K containing generators of H/K are equal. We denote by EG(H/K) the stabilizer of any q-cyclotomic class of H/K containing generators of H/K under the action of N. Further, the action of the previous paragraph induces an action of N on C(H/K). Let hK be a generator of H/K. The stabilizer of any element of C(H/K) equals

q qo−1 g qj StabN ({χ, χ , . . . , χ }) = {g ∈ N | χ = χ for some j} = {g ∈ N | χg(hK) = χ(hK)qj for some j} = {g ∈ N | χ((hK)g−1 ) = χ(hK)qj for some j} = {g ∈ N | hg−1 K = hqj K for some j}

= EG(H/K).

Note that we use the faithfulness of χ in the fourth step. Now we prove a lemma similar to Lemma 2.15.

Lemma 3.4. Let K ¡ H ≤ G be such that H/K is cyclic and let C ∈ C(H/K). 1. The following are equivalent for every g ∈ G:

(a) g ∈ K,

(b) gC (H,K) = C (H,K), e (c) gC (H,K) = C (H,K).

3.2. The primitive central idempotents 68 Chapter 3. Primitive central idempotents of finite group algebras

2. If H ¡ NG(K), then CenG(C (H,K)) = EG(H/K).

Proof. Let m = [H : K], o = om, = C (H,K) and ﬁx χ ∈ C and h ∈ H such that its image hK is a generator of H/K. Since χ is a generator of (H/K)∗, the image of χ is the set of m-th roots of unity in F(ξm) = Fqo and hence this image generates Fqo as F-vector space. We prove1. It is easy to see that 1(a) implies 1(b) and that 1(b) and 1(c) are equivalent. By Proposition 3.2, C (H/K, 1) is a primitive central idempotent of F(H/K) and therefore it is not zero. As a consequence, we have that 6= 0. Assume that g = . Then supp(g) 6= ∅ and g = ∈ FH, so g ∈ H. Assume now that g = kht, with k ∈ K and 0 ≤ t < m. Then

mX−1 mX−1 [H : K]−1Kf tr(χ(h−jK))ht+j = g = = [H : K]−1Kf tr(χ(h−jK))hj j=0 j=0 and hence tr(χ(ht+jK)) = tr(χ(hjK)) for every 0 ≤ j < m. Thus, tr((χ(htK) − 1)χ(hjK)) = 0 and hence χ(htK) = 1 because the image of χ contains a generating set of Fqo as F-vector space and tr : Fqo → F is F-linear and surjective. Thus, χ(hK)t = 1 and hence m | t because χ(hK) is a primitive mth root of unity. Therefore, g ∈ K. It remains to prove2. Using1, it is easy to show that Cen G() ≤ NG(K). Note that also EG(H/K) ≤ NG(K). Let g ∈ NG(K). By Proposition 3.2, C (H,K) and Cg (H,K) are two primitive central idempotents of FH and they are equal if and only if C = Cg, i.e. if g g ∈ EG(H/K). By equation (3.1), C (H,K) = Cg (H,K). We conclude that C (H,K) = g C (H,K) if and only if g ∈ EG(H/K), as desired. 2

Now, let p be the prime divisor of q and let Z(p) denote the localization of Z at p. Notice that we can identify Fp with the residue ﬁeld Z(p)/Z(p)p of Z(p). We denote the image of x ∈ Z(p) in Fp with x and extend this notation to the projection of Z(p)G onto FpG. Remember we have deﬁned (G, N) as an element of QG, for N ¡ G. But in fact, (G, N) belongs to Z(p)G and hence so does e(G, H, K) for K ¡ H ≤ G. Therefore, (G, N) is an idempotent of FpG and, if (H,K) is a strong Shoda pair, then e(G, H, K) is a central f idempotent of FpG. Note that (G, N) can be computed as in (2.2), but interpreting N and Ý the M’s as elements in FpG.

Lemma 3.5. 1. Let N ¡ G such that G/N is cyclic, then X (G, N) = C (G, N). C∈C(G/N)

2. Let K ≤ H ¡ NG(K) such that H/K is cyclic and R a set of representatives of the action of NG(K) on C(H/K). Then X e(G, H, K) = eC (G, H, K). C∈R

f Proof. We ﬁrst prove1. Since (G, N) and C (G, N) belong to FGN, for every C ∈ C(G/N), f by factoring out by N and using the isomorphism FGN ' F(G/N), we may assume without

3.2. The primitive central idempotents 69 Chapter 3. Primitive central idempotents of finite group algebras loss of generality that N = 1 and hence G is cyclic. By Proposition 3.2, every primitive central idempotent of FG is of the form C (G, H) with H ¡ G and C ∈ C(G/H). Therefore (G, 1) is the sum of some of the C (G, H). Thus, to prove1, it is enough to show that if H ¡ G and C ∈ C(G/H), then (G, 1)C (G, H) 6= 0 if and only if H = 1. If C ∈ C(G) and e e 1 6= x ∈ G, then (1 − x)C (G, 1) 6= 0, by Lemma 3.4, and hence (1 − x)C (G, 1) = C (G, 1) because C (G, 1) is a primitive idempotent. Since (G, 1) is a product of elements of the form e (1 − x) with 1 6= x ∈ G, we deduce that (G, 1)C (G, 1) = C (G, 1) 6= 0. On the other hand, if 1 6= H ≤ G, then there is h ∈ H such that M = hhi is a minimal nontrivial subgroup of e G and hence (G, 1)C (G, H) = (G, 1)(1 − h)C (G, H) = 0, by Lemma 3.4. This ﬁnishes the proof of the claim. We prove2. Let N = NG(K) = NG(H)∩NG(K), E = EG(H/K), TN a right transversal of N in G and TE a right transversal of E in N. Thus {hg | h ∈ TE, g ∈ TN } is a rightP transversal g of E in G. By Lemma 2.15, N = CenG((H,K)) and hence e(G, H, K) = g∈TN (H,K) . h We also know that C(H/K) is the disjoint union of the sets of the form {C | h ∈ TE} for C running on R and hence, using 3.1, one has

X g e(G, H, K) = (H,K) g∈T XN X g = C (H,K) g∈T C∈C(H/K) XN X X g = Ch (H,K) g∈T C∈R XN X hX∈TE hg = C (H,K) C∈R g∈T X N h∈TE = eC (G, H, K). C∈R 2

Now we can prove the main result of this chapter.

Theorem 3.6. Let G be a finite group and F a finite field of order q such that FG is semisimple.

1. Let (H,K) be a strong Shoda pair of G and C ∈ C(H/K). Then eC (G, H, K) is a primitive central idempotent of FG and

FGeC (G, H, K) ' M[G:H](Fqo/[E:H] ),

where E = EG(H/K) and o is the multiplicative order of q modulo [H : K].

2. Let X be a set of strong Shoda pairs of G. If every primitive central idempotent of QG is of the form e(G, H, K) for (H,K) ∈ X, then every primitive central idempotent of FG is of the form eC (G, H, K) for (H,K) ∈ X and C ∈ C(H/K).

Proof. 1. Set = C (H,K), e = eC (G, H, K) and let T be a right transversal of E = EG(H/K) in G. By LemmaP 3.4, E = CenG() and by Lemma 2.15, N = NG(K) = g CenG((H,K)). Thus e = g∈T .

3.2. The primitive central idempotents 70 Chapter 3. Primitive central idempotents of finite group algebras

Notice that e is central in FG and g is a primitive central idempotent in FHg for g ∈ T . Therefore, it is enough to show that the G-conjugates of are orthogonal. To prove this, it is enough to show that if g ∈ G \ E then g = 0. By Lemma 3.5, g = e(H,K) e(H,K)gg. By the definition of strong Shoda pair, if g∈ / N, then e(H,K)e(H,K)g = 0 and so g = 0. g g If g ∈ N \ E, then = Cg (H,K) and by Corollary 3.3, and are two different primitive central idempotents of FH. Thus, g = 0. By Corollary 3.3, FH is isomorphic to F(ξk) = Fqo , where k = [H : K]. Furthermore, σ FE = FH ∗τ E/H is a crossed product of E/H over the field FH, with homogeneous basis φ(E/H), where φ : E/H → E/K is a left invers of the canonical projection E/K → E/H. The action σ and twisting τ are given by the following formulas

−1 σ : E/H → Aut(FH): eH 7→ (α 7→ φ(eH) αφ(eH)),

−1 τ : E/H × E/H → U(FH):(e1H, e2H) 7→ φ(e1e2H) φ(e1H)φ(e2H). σ The isomorphism FH ' F(ξk) extends to an E/H-graded isomorphism FE = FH∗τ E/H ' σ1 F(ξk)∗τ1 E/H. Since H/K is maximal abelian in N/K and hence also in E/K, the action σ1 is faithful and hence FE is simple, by Lemma 1.32. Let Fqt be the fix subfield of the action σ1 on σ1 F(ξk), then one can consider the crossed product F(ξk)∗τ1 E/H as the classical crossed product o o (F(ξk)/Fqt , τ1). Moreover, E/H ' Gal(F(ξk)/Fqt ) and t = [Fq : Fqt ] = [E : H]. Since every finite division ring is a field, we have, by the Wedderburn Theorem, that FE ' M[E:H](Fqt ). If g ∈ G, then the map x 7→ xg is an isomorphism between the FG-modules FG and FGg. g [G:E] t Therefore, FGFGe = ⊕g∈T FG ' (FG) . Moreover FG = ⊕t∈T FEt = ⊕t∈T FEt = FE, because is central in FE and t = 0, for every t ∈ G \ E. Thus,

FGe ' EndFG(FGe) ' M[G:E](EndFG(FG)) ' M[G:E](FG) = M[G:E](FE) ' M[G:H](Fqo/[E:H] ). 2. By the assumption, there is a subset Y of X such that {e(G, H, K) | (H,K) ∈ Y } is the set of primitive central idempotents of QG. Then {e(G, H, K) | (H,K) ∈ Y } is a complete set of nonnecessarily primitive orthogonal central idempotents of FG. By1 and Lemma 3.5, {eC (G, H, K) | (H,K) ∈ Y,C ∈ R(H,K)} is the set of primitive central idempotents of FG, where R(H,K) denotes a set of representatives of the orbits of the action of NG(K) on C(H/K). 2

Applying the results of Chapter2, one obtain the following.

Corollary 3.7. If G is an abelian-by-supersolvable group and F is a ﬁnite ﬁeld of order q such that FG is semisimple, then every primitive central idempotent of FG is of the form eC (G, H, K) for (H,K) a strong Shoda pair of G and C ∈ C(H/K). Furthermore, for every strong Shoda pair (H,K) of G and every C ∈ C(H/K), FGeC (G, H, K) ' M[G:H](Fqo/[E:K] ), where E = EG(H/K) and o is the multiplicative order of q modulo [H : K].

Corollary 3.8. Let G be a finite metabelian group, A a maximal abelian subgroup of G containing G0 and F a finite field of order q such that FG is semisimple. Then every primitive central idempotent of FG is of the form eC (G, H, K) for (H,K) a pair of subgroups of G satisfying the following conditions

3.2. The primitive central idempotents 71 Chapter 3. Primitive central idempotents of finite group algebras

1. H is a maximal element in the set {B ≤ G | A ≤ B and B0 ≤ K ≤ B};

2. H/K is cyclic; and C ∈ C(H/K). Furthermore, for every pair (H,K) of subgroups of G satisfying1 and2 and every C ∈ C(H/K), FGeC (G, H, K) ' M[G:H](Fqo/[E:K] ), where E = EG(H/K) and o is the multiplicative order of q modulo [H : K].

3.3 Examples

In Chapter2, we explained algorithms to compute the strong Shoda pairs of a group G. We also had algorithms to compute the primitive central idempotent of QG realizable by a strong Shoda pair and to compute the simple algebra of QG associated to a strong Shoda pair. Using Theorem 3.6, one can write the algorithms for the ﬁnite case. Actually, this is already done and implemented in the package “Wedderga” [CKO+09]. We will illustrate this with some examples.

Example 3.9. Consider the quaternion group of order 8, Q8 = hx, yi. This group is abelian- by-supersolvable and therefore we can compute the Wedderburn decomposition of FqG, for each (q, 8) = 1. In this example we compute F3Q8. gap> F:=FreeGroup("x","y");;x:=F.1;;y:=F.2;; gap> G:=F/[x^4,x^2*y^2,y^(-1)*x*y*x];; gap> FG:=GroupRing(GF(3),G);; gap> IsSemisimpleFiniteGroupAlgebra(FG); true gap> PrimitiveCentralIdempotentsByStrongSP(FG); [ (Z(3))*+(Z(3))*x+(Z(3))*y+(Z(3))*x^2+(Z(3))*x*y+(Z(3))*x^ 3+(Z(3))*x^2*y+(Z(3))*x^3*y, (Z(3))*+(Z(3)^0)*x+(Z(3)^ 0)*x*y^-1*x^-1+(Z(3))*x^2+(Z(3))*x*y^-1*x^2+(Z(3)^0)*x^-1+(Z(3)^0)*x*y^ -1*x+(Z(3))*x*y^-1, (Z(3))*+(Z(3)^0)*x+(Z(3))*y^-1*x^2+( Z(3))*x^2+(Z(3)^0)*y^-1*x+(Z(3)^0)*x^-1+(Z(3))*y^-1+(Z(3)^0)*y^-1*x^-1, (Z(3))*+(Z(3))*x^-3+(Z(3)^0)*x^-2*y^-1+(Z(3))*x^-2+(Z(3)^ 0)*x^-1*y^-1+(Z(3))*x^-1+(Z(3)^0)*y^-1+(Z(3)^0)*x^-3*y^-1, (Z(3))*+(Z(3)^0)*x^6 ] gap> WedderburnDecompositionInfo(FG); [[1,3],[1,3],[1,3],[1,3],[2,3]]

Ý 2 GAP tells us there are 5 primitive central idempotents, namely Q8, (2 + x)(1 + 2y + x + 2x2y), (2 + x)(1 + y + x2 + x2y), 2(1 + x)(1 + 2y + x2 + 2x2y) and 2 + x2. The GAP output also tells us there are 5 simple components, each of the form [n, k]. This means, each component is of the form Mn(Fk). We conclude that

F3Q8 ' 4F3 ⊕ M2(F3).

In Example 2.39, we saw that QQ8 is not isomorphic to QD8. We will see that this is no longer the case for ﬁnite group algebras.

4 2 −1 Example 3.10. Let D8 = ha, b | a = 1, b = 1, bab = a i be the dihedral group of order 8. We compute the Wedderburn decomposition for F3D8.

3.3. Examples 72 Chapter 3. Primitive central idempotents of finite group algebras

gap> F:=FreeGroup("a","b");;a:=F.1;;b:=F.2;; gap> G:=F/[a^4,b^2,b*a*b*a];; gap> FG:=GroupRing(GF(3),G);; gap> WedderburnDecompositionInfo(FG); [[1,3],[1,3],[1,3],[1,3],[2,3]]

We see that F3D8 ' 4F3 ⊕ M2(F3).

Hence F3Q8 is isomorphic to F3D8.

Example 3.11. Consider the dihedral group of order 16, D16 and the ﬁeld of order 5, F5. Note that (5, 16) = 1 and therefore F5D16 is semisimple. We compute the Wedderburn decomposition of F5D16. gap> G:=DihedralGroup(16);; gap> FG:=GroupRing(GF(5),G);; gap> WedderburnDecompositionInfo(FG); [[1,5],[1,5],[1,5],[1,5],[2,5],[2, 25 ]]

We conclude that F5D16 ' 4F5 ⊕ M2(F5) ⊕ M2(F52 ).

3.3. Examples 73 Chapter 4

A complete set of orthogonal primitive idempotents

In Chapter2, we proved that, for G an abelian-by-supersolvable group, the primitive central idempotents of QG are of the form e(G, H, K), with (H,K) a strong Shoda pair of G and we described the simple component QGe(G, H, K). In Chapter3, we proved a similar result for finite group algebras. We will use this in order to study the primitive idempotents of group algebras for finite nilpotent groups. Note that we can do this because a finite nilpotent group is abelian-by-supersolvable.

4.1 The rational group ring of nilpotent groups

In this section we will describe a complete set of orthogonal primitive idempotents and moreover, a complete set of matrix units of QGe(G, H, K), with G a ﬁnite nilpotent group and (H,K) a strong Shoda pair of G. This section is based on a recent article of Jespers, Olteanu and del R´ıo[JOdR10]. We will ﬁrst produce a complete set of orthogonal primitive idempotents of a classical crossed product with trivial twisting. By Corollary 1.34 the classical crossed product (K/F, 1) is isomorphic to Mn(F ), with n = [K : F ]. Therefore, a complete set of orthogonal primitive idempotents of (K/F, 1) contains n idempotents.

Lemma 4.1. Let K be a ﬁeld of characteristic zero and let A = (K/F, 1) be a classicalP 1 crossed product with trivial twisting and let G = Gal(K/F ) with |G| = n. Let e = |G| g∈G g and let x1, . . . , xn be non-zero elements of K. Then the conjugates of e by x1, . . . , xn form −1 a complete set of orthogonal primitive idempotents of A if and only if trK/F (xixj ) = 0 for every i 6= j. Proof. As the twisting is trivial, {g | g ∈ G} is a subgroup of order |G| of the group of units of A and hence e is an idempotent ofA. NoteP that the conjugates of e by non-zero elements of L are non-zero. Moreover, ge = g−1 1 gh = e for every g ∈ G, since ⊆ F . Therefore, P |G| h∈GP P Q 1 1 −1 1 1 if x ∈ K, then exe = |G| g∈G gxe = |G| g∈G g (x)ge = |G| g∈G g(x)e = |G| trK/F (x)e. −1 −1 −1 −1 Thus if xi, xj ∈ K, i 6= j, then xi exi and xj exj are orthogonal if and only of xi exixj e = 0 −1 −1 −1 if and only if xi trK/F (xixj )e = 0 if and only if trK/F (xixj ) = 0. Since we have found n orthogonal idempotents of A, these form a complete set of orthogonal primitive idempotents

74 Chapter 4. A complete set of orthogonal primitive idempotents of A. 2

In the proof of Theorem 4.7, we will encounter some examples of classical crossed products with trivial twisting with a list x1, . . . , xn satisfying the conditions of Lemma 4.1. We give another example.

Example 4.2. Let K/F be a cyclic extension of order n, that is a Galois extension K/F whose Galois group is cyclic of order n. Assume F contains a primitive n-th root of unity. By a Theorem of Kummer, we have that K = F [α] for a α ∈ K with αn ∈ F . Let a = αn, n−1 then x1 = 1, x2 = α, . . . , xn = α satisfy the conditions of Lemma 4.1. Indeed, the minimal i n/d i/d n/d i n/d polynomial of α over F for 1 ≤ i < n is of the form fi(X) = X − a = X − (α ) i i i for d = gcd(n, i). Therefore trK/F (α ) = [K : F (α )]trF (αi)/F (α ). Since the degree of fi n i i i is d = [F (α ): F ], trF (αi)/F (α ) is the sum of all roots of fi. Therefore trF (αi)/F (α ) = i i 2 i n/d−1 i n α + ξn/dα + ξn/dα + ··· + ξn/d α , with ξn/d a primitive d -th root of unity. Therefore i i −i trF (αi)/F (α ) = 0 and also trK/F (α ) = 0. Similarly trK/F (α ) = 0. However there are cases where a classical crossed product with trivial twisting does not have elements x1, . . . , xn satisfying the conditions of Lemma 4.1. This is the case in the next Example.

Example 4.3. Consider the ﬁeld K = Q(ξ7). Then K/Q is a cyclic Galois extension of degree 6. Let H be the unique subgroup of Gal(K/Q) of order 3, i.e. H is generated by 2 2 4 2 √ξ7 7→ ξ7 . Denote F √= Fix(H). Let β = ξ7 + ξ7 + ξ7 ∈ F√. Then β + β + 2 = 0 and√ hence −7 ∈ F . Since [Q( −7) : Q] = 2, we know that [K : Q( −7)] = 3 and hence F = Q( −7), since also [K : F ] = 3. Now consider the classical crossed product (K/F, 1). Let σ be the −1 generator of Gal(K/F ). If x1, x2, x3 satisfy the conditions of Lemma 4.1, then α = x2x1 is a non-zero element of K with zero trace over F . Then (X − α)(X − σ(α))(X − σ2(α)) = 3 2 2 2 2 X − trK/F (α)X + (ασ (α) + σ(α)σ (α) + ασ(α))X − ασ(α)σ (α). Since trK/F (α) = 0 = ασ2(α) + σ(α)σ2(α) + ασ(α), the minimal polynomial of α over F is of the form X3 − a for some a ∈ F . Since the degree of f(X) = X3 − a equals |Gal(K/F )|, f splits over K and hence ξ3α ∈ K. But this implies that K contains a 3th root of unity, a contradiction. To classify the ﬁnite p-groups G with a maximal abelian subgroup which is cyclic and normal in G, we need to following Lemma.

Lemma 4.4. Let A be a cyclic group of order pn, with p prime. If either p is odd or p = 2 and n ≤ 2 then Aut(A) is cyclic. Otherwise Aut(A) = hφ5i×hφ−1i, where φr is the automorphism r of A given by φr(a) = a .

Proof. Remember there is an isomorphism Aut(A) = {φr | (r, p) = 1}'U(Zpn ), given by φr 7→ r. We will prove that U(Zpn ) is cyclic if p is odd or p = 2 and n ≤ 2, and that n n−1 U(Z2n ) = h5i × h−1i if n > 2. Note that |U(Zpn )| = φ(p ) = p (p − 1). First consider the case when p is odd. If n = 1, then it is well known that U(Zp) is cyclic. We claim that U(Zp2 ) is cyclic. Suppose z is a generator of U(Zp) and denote o the order 2 o 2 o of z in U(Zp2 ). Then o is a divisor of φ(p ) = p(p − 1). Also, if z ≡ 1 mod p , then z ≡ 1 mod p. Therefore p−1 divides o. It follows that either o = p−1 or either o = p(p−1). In the

4.1. The rational group ring of nilpotent groups 75 Chapter 4. A complete set of orthogonal primitive idempotents

latter case we have found a generator of U(Zp2 ). Now suppose that o = p − 1 and note that z + p is still of order p − 1 in U(Zp). By the previous arguments, the order of z + p in U(Zp2 ) is either p − 1 or p(p − 1). We will prove that the first case cannot occur. It is easy to see that (z + p)p−1 ≡ zp−1 − zp−2p mod p2. Since z has order p − 1, we get that (z + p)p−1 ≡ 1 − zp−2p mod p2. Since (p, z) = 1, we get that (z + p)p−1 6≡ 1 mod p2. Next we claim that a generator of U(Zp2 ) is also a generator of U(Zpm ), for m ≥ 2. Let z be a generator of U(Zp2 ), this means that the order of z is p(p − 1). Then denote the order of m m−1 o z in U(Zpm ) by o. This means that o is a divisor of φ(p ) = p p. Moreover, since z ≡ 1 mod pm, also zo ≡ 1 mod p2. Therefore, p(p − 1) | o. This implies that o = pk(p − 1) for some 1 ≤ k ≤ m − 1. We will prove that k = m − 1. To do this, it is sufficient to show that zpm−2(p−1) 6≡ 1 mod pm. We will do this by induction on m. When m = 2, then zp−1 6≡ 1 mod p2 since z is of order p(p − 1). Now assume we know the result for m − 1. We know that zφ(pm−2) ≡ 1 mod pm−2. Hence, zpm−3(p−1) = 1 + cpm−2 for some c. We know that p is not a divisor of c, for otherwise zpm−3(p−1) ≡ 1 mod pm−1, contradicting the induction hypothesis. m−2 We conclude that zp (p−1) = (1 + cpm−2)p ≡ 1 + cpm−1 mod pm. Since p - c, we have that zpm−2(p−1) 6≡ 1 mod pm. This proves the case when p is odd. When p = 2 and n ≤ 2, it is easy to see that U(Z2n ) is cyclic. Now assume p = 2 and n ≥ 3. Since −1 has order 2, it suffices to prove that 5 has order 2n−2 modulo 2n and 5m 6≡ −1 mod 2n for m ∈ {1,..., 2n−2}. We will do this by induction. When n = 3, it is easy to see that 5 has order 2 modulo 8 and that 5m 6≡ −1 mod 8. Now assume we know the result for n − 1. Denote the order of 5 modulo 2n by o. Then is o a divisor of φ(2n) = 2n−1. Since 5o ≡ 1 mod 2n, 5o ≡ 1 mod 2n−1 and therefore 2n−3 is a divisor of o. If we show that 52n−3 6≡ 1 mod 2n and 52n−2 ≡ 1 mod 2n, the order of 5 modulo 2n is 2n−2. By induction we have that 52n−3 ≡ 1 mod 2n−1. Hence 52n−3 = 1 + c2n−1 for some c. Therefore 52n−2 = 1 + c2n + c222(n−1) ≡ 1 mod 2n. By the induction hypothesis, we have that 52n−4 6≡ 1 mod 2n−1 and 52n−4 ≡ 1 mod 2n−2 and therefore, 52n−4 = 1 + c2n−2, for some c relatively prime to 2. Hence 52n−3 = 1 + c2n−1 mod 2n, and 52n−3 6≡ 1 mod 2n. Also, if there exist m ∈ {1,..., 2n−2} such that 5m ≡ −1 mod 2n, then also 5m ≡ −1 mod 2n−1. By the induction hypothesis, it follows that m = 2n−2. This implies that 1 ≡ −1 mod 2n, a contradiction. 2

The groups listed in the following Lemma will be the building blocks in the proof of Theorem 4.7. For n and p integers with p prime, we use vp(n) to denote the valuation at p of n, i.e. pvp(n) is the maximal p-th power dividing n.

Lemma 4.5. Let G be a ﬁnite p-group which has a maximal abelian subgroup which is cyclic and normal in G. Then G is isomorphic to one of the groups given by the following presentations:

pn pk −1 r P1 = ha, b | a = b = 1, b ab = a i,

with either vp(r − 1) = n − k, or p = 2 and r 6≡ 1 mod 4, or k = 0, 2n 2k 2 −1 r −1 −1 P2 = ha, b, c | a = b = c = 1, bc = cb, b ab = a , c ac = a i, with r ≡ 1 mod 4, 2n 2k 2 2n−1 −1 r −1 −1 P3 = ha, b, c | a = b = 1, c = a , bc = cb, b ab = a , c ac = a i, with r ≡ 1 mod 4.

4.1. The rational group ring of nilpotent groups 76 Chapter 4. A complete set of orthogonal primitive idempotents

Note that if k = 0 (equivalently, if b = 1), then the ﬁrst case corresponds to the case when G is abelian (and hence cyclic), the second case coincides with the ﬁrst case with p = 2, k = 1 and r = −1, and the third case is the quaternion group of order 2n+1. Proof. Let A be a maximal abelian subgroup of G and assume A = hai is cyclic and normal n in G. Assume |A| = p . If G is abelian, then G = A. Hence G ' P1 with k = 0. Assume G is not abelian, or equivalently, A ( G. Consider the action of G on A by inner automorphisms. Since A is maximal abelian in G, the kernel of this action is A and therefore G/A is isomorphic to a subgroup of Aut(A). By Lemma 4.4, Aut(A) is cyclic if either p is odd or p = 2 and n ≤ 2. Otherwise, Aut(A) = hφ5i × hφ−1i. Assume that G/A is cyclic, then G is metacyclic, so G has a presentation of the form

ha, b | apn = 1, bpk = as, b−1ab = ari, (4.1) with pn | s(r − 1) and pn | rpk − 1. We know that 22i ≥ 2i+2 if i ≥ 2 (or 2i ≥ 3) and that if p is odd and j ≥ 2, then p1+ij ≥ pi+2 and pip ≥ pi+2 if i ≥ 1 (or pi ≥ 3). Therefore, if pi ≥ 3 then p(p − 1) (1 + xpi)p = 1 + xpi+1 + x2p2i + ··· + xppip ≡ 1 + xpi+1 mod pi+2, (4.2) 2

i−1 pi−1 for every i ≥ 1 and x ∈ Z. Let i ≥ 1 and assume that rp − 1 = jpvp(r −1), with (j, p) = 1. i pi−1 pi−1 pi−1 pi−1 Then rp = (1 + jpvp(r −1))p ≡ 1 + jpvp(r −1)+1 mod pvp(r −1)+2, if pvp(r −1) ≥ 3. i pi−1 pi−1 Therefore rp − 1 = (j + pk)pvp(r −1)+1, for some k ∈ Z, if pvp(r −1) ≥ 3. If p is odd, then pi−1 k always pvp(r −1) ≥ 3 since p | rp −1 and by induction (using formula (4.2) and the equation i−1 2i−1 i−1 p p p v2(r −1) 2 (x − 1) ≡ x − 1 mod p) p | r − 1. If p = 2, then 2 ≥ 3 if v2(r − 1) ≥ 2. This is true if r2i−1 − 1 is divisible by 4, so by induction if r − 1 is divisible by 4. Hence, if either p is odd or p = 2 and r ≡ 1 mod 4, then

pi pi−1 vp(r − 1) = vp(r − 1) + 1, (4.3) for every i ≥ 1. Furthermore, since pn | rpk − 1 and A is maximal abelian in G, one deduces pk pk−1 pk−1 that n ≤ vp(r − 1) = vp(r − 1) + 1 ≤ n, for otherwise ha, b i is abelian. Hence, pk vp(r − 1) = n and vp(r − 1) = n − k.

k n rp −1 Since p | s(r − 1), we conclude that vp(s) ≥ n − vp(r − 1) = k = vp( r−1 ) = vp(1 + r + k k 2 p −1 i1 2 p −1 i2 r + ··· + r ). Hence, s = p q1 and 1 + r + r + ··· + r = p q2 for some i1 ≥ i2 and i1−i2 some q1 and q2, with (p, qi) = 1. Since the equation xq2 + p q1 = 0 is solvable in Zpn−i2 in x, there exists an integer x such that x(1 + r + r2 + ··· + rpk−1) + s ≡ 0 mod pn. Then k k (bax)pk = bpk ax(1+r+r2+···+rp −1) = as+x(1+r+r2+···+rp −1) = 1, and, replacing b by bax in (4.1), we obtain the presentation of P1. We have also proved that vp(r − 1) = n − k unless p = 2 and r 6≡ 1 mod 4. Assume that p = 2 and r 6≡ 1 mod 4, then v2(r − 1) = 1, since 2 | r − 1, k 2 2k−1 r2 −1 so v2(s) ≥ n − v2(r − 1) = n − 1 and v2(1 + r + r + ··· + r ) = v2( r−1 ) ≥ n − 1. s We consider 3 cases. First consider the case when v2(s) ≥ n, then a = 1 and therefore 2 2k−1 2k G ' P1. Secondly, assume that v2(s) = v2(1 + r + r + ··· + r ) = n − 1, then (ba) = k 2k−1 2k−1 n−1 k 2 1+r+···+r s+1+r+···+r 2 (q1+q2) 2 b a = a = a , with both q1 and q2 odd, hence (ba) = 1.

4.1. The rational group ring of nilpotent groups 77 Chapter 4. A complete set of orthogonal primitive idempotents

Replacing b by ba in (4.1), we obtain again that G ' P1. Last, assume that v2(s) = n − 1 2k−1 2k 2k−1 and v2(1 + r + ··· + r ) ≥ n. Then, v2(r − 1) = v2(1 + r + ··· + r ) + 1 > n and 2k−1 2k 2k−1 n > v2(r − 1) by the maximality of A. But then, v2(r − 1) > v2(r − 1) + 1. Also note that, by induction, r2m − 1 is divisible by 4 for m ≥ 1. Assume now that k ≥ 2, then 2k−1 2k 2k−1 v2(r −1) ≥ 2 and hence, by (4.2), v2(r −1) = v2(r −1)+1, a contradiction. Therefore, k = 1. We have that s = 2n−1q, with (2, q) = 1 and r = 1 + 2q0 with (2, q0) = 1. Thus G has a presentation haq, b | (aq)2n = 1, b2 = (aq)2n−1 , b−1aqb = (aq)ri, and by replacing aq by a, we have a presentation

ha, b | a2n = 1, b2 = a2n−1 , b−1ab = ari.

Remember that 2n | r2 − 1 and hence r2 = 1 or n ≤ 2. Note that r and n cannot be 1 since A is maximal abelian in G. In both cases ar = a−1 and hence we obtain the presentation

ha, b | a2n = 1, b2 = a2n−1 , b−1ab = a−1i,

n+1 the quaternion group of order 2 which is P3 for k = 0. Assume now that G/A is not cyclic, so p = 2 and G/A = hbi × hci, with c−1ac = a−1 and b−1ab = ar, with r = 5m for some positive integer m, so r ≡ 1 mod 4. Also, c2 = aj, for some j. Since c2 = aj commutes with c, aj has to be of order 2. The only elements of order 2 are 1 and a2n−1 . Therefore G has a presentation of the form

ha, b, c | a2n = 1, b2k = as, c−1ac = a−1, cb = aibc, b−1ab = ar, c2 = 1 or c2 = a2n−1 i, (4.4) with 2n | s(r − 1) and 2n | r2k − 1. Since r ≡ 1 mod 4, equation (4.3) holds also in this 2k case and therefore v2(r − 1) = n and v2(r − 1) = n − k ≥ 2. Take an integer x such that x(1+r +r2 +···+r2k−1)+s ≡ 0 mod 2n, and replace b by bax in (4.4), then one may assume k k 2k 2k i(1+r+···+r2 −1) n 2k−1 r2 −1 that b = 1. Then c = cb = a c and so 2 | i(1 + r + ··· + r ) = i r−1 . As k r2 −1 v2( r−1 ) = k, we have v2(ir) = v2(i) ≥ n − k = v2(r − 1). Hence, there exists an integer j such that j(r − 1) + ir ≡ 0 mod 2n. It is easy to verify that the commutator of b and ajc is 1. So, replacing c by ajc if needed, we may assume that b and c commute and we obtain the 2 2 2n−1 presentation of P2 if c = 1, and the presentation of P3, if c = a . 2 −1,−1 We also need the following result on a Hamiltonian quaternion algebra H(F ) = F .

Lemma 4.6. Let F be a ﬁeld of characteristic zero. Then the quaternion algebra H(F ) splits 2 2 1 1 if and only if x + y = −1 for some x, y ∈ F . In that case 2 (1 + xi + yj) and 2 (1 − xi − yj) form a complete set of orthogonal primitive idempotents of H(F ). −1 Furthermore, if F = Q(ξm, ξ2n + ξ2n ) with m odd, then −1 is the sum of two squares of F if and only if m 6= 1 and either n ≥ 3 or the multiplicative order of 2 modulo m is even. 1 Proof. The ﬁrst part is a consequence of Proposition 1.37. It is easy to verify that 2 (1 + 1 xi + yj) and 2 (1 − xi − yj) are orthogonal idempotents of H(F ) and their sum is 1. Since H(F ) ' M2(F ), a complete set of orthogonal primitive idempotents can only contain 2 elements and therefore we have found such a set. −1 Now assume that F = Q(ξm, ξ2n + ξ2n ) with m odd. If m = 1, then F is totally real and therefore −1 is not the sum of two squares of F . So assume that m 6= 1. If n ≤ 2,

4.1. The rational group ring of nilpotent groups 78 Chapter 4. A complete set of orthogonal primitive idempotents then F = Q(ξm) and −1 is a sum of squares in F if and only if the multiplicative order of 2 modulo m is even [MS02, Th. 7.4.7 (Moser)] or [Lam73, p307-308]. Finally, assume that m 6= 1 and n ≥ 3. By the Hasse principal [Lam73, p168-169], we have to show that −1 is a sum of squares in Fp for all places p. For the inﬁnite places, the result holds since F cannot be embedded in R since m 6= 1 and clearly, −1 is a sum of squares in C. For the odd places, the result follows from√ [Lam73, Ex. 2.4(6) p304]. We prove the fact for p = 2. Since n ≥ 3, −1 F contains√ξ8 + ξ8 = 2 and, as 2 is not a square in Q2 [Lam73, Cor. 2.24 p162], we deduce that [Q2( 2) : Q2] = 2 and hence [F2 : Q2] is even. By [Lam73, Ex. 2.4(7) p304], −1 is a sum of squares in F2. This completes the proof. 2

Now we are ready to show a method to calculate a complete set of orthogonal primitive idempotents of QG for G a ﬁnite nilpotent group. Since G is abelian-by-supersolvable, it follows from Theorem 2.25 that every primitive central idempotent of QG is of the form e(G, H, K) with (H,K) a strong Shoda pair of G and therefore it is enough to obtain a complete set of orthogonal primitive idempotents of QGe(G, H, K) for every strong Shoda pair (H,K) of G. This is described in the main result of this Chapter that we state now. 0 We will needQ the notions of 2-part and 2 -part of a group. Let G be a ﬁnite nilpotent group, then G = p Sp, the direct product of the Sylow p-subgroups Sp for p | |G|. The 2-partQ of a nilpotent group G is the Sylow 2-subgroup S2. The 2’-part is the direct product p6=2 Sp, of all Sylow p-subgroups with p 6= 2.

Theorem 4.7. Let G be a ﬁnite nilpotent group and (H,K) a strong Shoda pair of G. Set e = e(G, H, K), = (H,K), H/K = hai, N = NG(K) and let N2/K and H2/K = ha2i (respectively N20 /K and H20 /K = ha20 i) denote the 2-parts (respectively 2’-parts) of N/K and H/K respectively. Then ha20 i has a cyclic complement hb20 i in N20 /K. A complete set of orthogonal primitive idempotents of QGe consists of the conjugates of Ý 2 [N20 :H20 ]−1 b20 β2 by the elements of T20 T2TG/N , where T20 = {1, a20 , a20 , . . . , a20 }, TG/N denotes a right transversal of N in G and β2 and T2 are given according to the cases below. g (1) If H2/K has a complement M2/K in N2/K then β2 = M2. Moreover, if M2/K is cyclic, then there exists b2 ∈ N2 such that N2/K is given by the following presentation

n k 2 2 b2 r ha2, b2 | a2 = b2 = 1, a2 = a2 i,

and if M2/K is not cyclic, then there exist b2, c2 ∈ N2 such that N2/K is given by the following presentation

n k 2 2 2 b2 r c2 −1 ha2, b2, c2 | a2 = b2 = 1, c2 = 1, a2 = a2 , a2 = a2 , [b2, c2] = 1i,

2n−2 with r ≡ 1 mod 4 (or equivalently a2 is central in N2/K.) Then

2 2k−1 2n−2 (i) T2 = {1, a2, a2, . . . , a2 }, if a2 is central in N2/K (unless n ≤ 1) and M2/K is cyclic; and n−2 2 [N2:H2]/2−1 2n−2 2n−2+1 2 +[N2:H2]/2−1 (ii) T2 = {1, a2, a2, . . . , a2 , a2 , a2 , . . . , a2 }, otherwise.

(2) If H2/K has no complement in N2/K, then there exist b2, c2 ∈ N2 such that N2/K is given by the following presentation

n k n−1 2 2 2 2 b2 r c2 −1 ha2, b2, c2 | a2 = b2 = 1, c2 = a2 , a2 = a2 , a2 = a2 , [b2, c2] = 1i,

4.1. The rational group ring of nilpotent groups 79 Chapter 4. A complete set of orthogonal primitive idempotents

with r ≡ 1 mod 4 and we set m = [H20 : K]/[N20 : H20 ]. Then

Ü 2 2k−1 (i) β2 = b2 and T2 = {1, a2, a2, . . . , a2 }, if either m = 1 or the order of 2 modulo m is odd and n − k ≤ 2 and n−2 n−2 2 2 k k Ü 1+xa2 +ya2 c2 2 2 −1 2 2 −1 (ii) β2 = b2 2 and T2 = {1, a2, a2, . . . , a2 , c2, c2a2, c2a2, . . . , c2a2 } with k k [N20 :H20 ] 2 −2 x, y ∈ Q a20 , a2 + a2 , satisfying (1 + x2 + y2) = 0, if m 6= 1 and either the order of 2 modulo m is even or n − k > 2.

Proof. Take T = TG/N a right transversal of N in G. Note that the conjugates of by T form a set of [G : N] distinct orthogonal idempotents whose sum is e, and therefore, we have t t0 the Pierce decomposition QGe = ⊕t,t0∈T QG . We also know from the proof of Theorem 2.17 that tQGt = tQNt = QNt ' QN. Hence the conjugates of by T are the “diagonal” elements in the matrix algebra QGe ' M[G:N](QN) = M[G:N](QH ∗ N/H) (Theorem 2.17). Hence, it is sufficient to compute a complete set of orthogonal primitive idempotents for QN = QH ∗ N/H and then add their T -conjugates in order to obtain the primitive idempotents of QGe. So, if we can prove that the orthogonal primitive idempotents of QN Ý are formed by conjugating b20 β2 by elements of T20 T2, then it follows that a complete set of Ý orthogonal primitive idempotents of QGe consists of the conjugates of b20 β2 by T20 T2TG/N . So one may assume that G = N, i.e. K is normal in G and hence e = and T = {1}. f Remember that the natural isomorphism QGK ' Q(G/K) maps to (H/K) and (H/K) = (H/K, 1). If we can show that a complete set of orthogonal primitive idem- Ý potents of Q(G/K)(H/K, 1) consists of the conjugates of b20 β2(H/K, 1) by T20 T2, then the Ý orthogonal primitive idempotents of QG are the conjugates of b20 β2 by T20 T2. So, we may assume that K = 1. Hence H = hai is a cyclic maximal abelian subgroup of G (SS2), which is normal in G (SS1) and e = = (H). The map a 7→ ξ|H| induces an isomorphism φ : QH ' Q(ξ|H|). If G = H, then QGe = QH ' Q(ξ|H|), a field. Then is the only nonzero idempotent. With the notations of the Ý Theorem, we are in case (1)(i) and β2 = 1 = b20 and T2 = {1} = T20 . Hence the result follows. So, in the remainder of the proof, we assume that G 6= H. Using the description of QGe given in Remark 2.18, one obtains a description of QGe as a classical crossed product (Q(ξ|H|)/F, τ), where F is the image under φ of the center of QGe. Note that Gal(Q(ξ|H|)/F ) ' G/H and hence [Q(ξ|H|): F ] = [G : H]. We first consider the case when G is a p-group. Then G and H = hai satisfy the conditions of Lemma 4.5 and therefore G is isomorphic to one of the three groups of this Lemma and n |H| = p . Moreover, H has a complement in G if and only if G ' P1 or G ' P2. In these cases, τ is trivial, since the complement of H in G is abelian. We claim that in these cases it is l possible to give a list of elements x , . . . , x l of (ξ )(p = [G : H]) satisfying the conditions 1 Pp Q |H| 1 Ý 0 of Lemma 4.1 and that the element [G:H] gH∈G/H gH corresponds to b2 β2 and the elements −1 −1 φ (x1), . . . , φ (xpl ) correspond to the conjugating elements in G given inP the statement of the Theorem in the different cases. In this case, the conjugates of 1 g [G:H] g∈Gal(Q(ξ|H|)/F ) by x1, . . . , xpl form a completeP set of orthogonal primitive idempotents of (Q(ξ|H|)/F, τ) and 1 −1 −1 hence the conjugates of [G:H] gH∈G/H gH by φ (x1), . . . , φ (xpl ) form a complete set of orthogonal primitive idempotents of QGe.

4.1. The rational group ring of nilpotent groups 80 Chapter 4. A complete set of orthogonal primitive idempotents

To prove this, we will use the following fact. If K is a subfield of Q(ξ|H|) such that ξp ∈ K, i l ±i ξ|H| 6∈ K for i = 1, . . . , p − 1 and, moreover ξ4 ∈ K if p = 2, then trQ(ξ|H|)/K (ξ|H|) = 0. To ipd see this notice that there exist a minimal integer d such that 1 6= ξ|H| ∈ K, since ξp ∈ K. i pd ipd Then ξ|H| is a root of X − ξ|H| ∈ K[X]. Note that Q(ξ|H|)/Q is an abelian Galois extension and Gal(Q(ξ|H|)/Q) is isomorphic to a subgroup of Aut(hξ|H|i), so by Lemma 4.4, it is cyclic or isomorphic to a subgroup of hφ5i × hφ−1i. As a consequence also Q(ξ|H|)/K is an abelian i Galois extension. By the Fundamental Theorem of Galois Theory, also Q(ξ|H|)/K is an abelian d i Galois extension of degree p . Moreover, since ξ4 ∈ K if p = 2, Gal(Q(ξ|H|)/K) is cyclic of d pd ipd i degree p , because φ−1(ξ4) 6= ξ4. Hence X −ξ|H| is the minimal polynomial of ξ|H| over K and i i i −i tr (ξ )/K (ξ ) = [ (ξ|H|): (ξ )]tr i (ξ ) = 0. Analogously, tr (ξ )/K (ξ ) = 0. Q |H| |H| Q Q |H| Q(ξ|H|)/K |H| Q |H| |H| Assume first that G ' P1 and vp(r − 1) = n − k ≥ 1 (that is, either p is odd, or p = 2 and r ≡ 1 mod 4). This implies that apk ∈ Z(G). Note that n ≥ 2 since G 6= H. Since k G/H ' Gal(Q(ξ|H|)/F ), F is a subfield of index [G : H] = p in Q(ξ|H|) and such that if p = 2, 2n−2 2n−2 pk then a ∈ Z(G) (since r ≡ 1 mod 4) and hence ξ4 = φ(a ) ∈ F . Since a ∈ Z(G), pk pk i ξ|H| = ξpn−k ∈ F and therefore, F = Q(ξ|H|) = Q(ξpn−k ). Moreover, ξp ∈ F and ξ|H| ∈/ F for k i k −1 i−j i = 1, . . . , p −1. If we set xi = ξ|H|, for i = 0, 1, . . . , p −1, then xixj = ξ|H| ∈/ F if i 6= j and −1 hence tr (x x ) = 0. Then the elements x , . . . , x k satisfy the conditions of Lemma Q(ξ|H|)/F i j P 0 p −1 e 1 −1 −1 −1 4.1, and hence the conjugates of b = [G:H] gH∈G/H gH by φ (x0), φ (x1) . . . , φ (xpk−1) e form a complete set of orthogonal primitive idempotents of QGe. If p = 2, then b20 = 1, β2 = b, −1 −1 2 −1 2k−1 −1 T20 = {1} and T2 = {1 = φ (x0), a = φ (x1), a = φ (x2), . . . , a = φ (x2k−1)}. If p −1 −1 2 −1 pk−1 −1 is odd, then b20 = b, T20 = {1 = φ (x0), a = φ (x1), a = φ (x2), . . . , a = φ (xpk−1)}, β2 = 1 and T2 = {1}. In both cases, this is case (1)(i) and we see that the statement of the Theorem holds. 2n−2 Assume now that G is still P1, but with p = 2 and r 6≡ 1 mod 4, or equivalently a k is not central. Then, ξ4 ∈/ F and F is of index [G : H] = 2 in Q(ξ|H|). Since, F has index k−1 2 in F (ξ4), F (ξ4) is a subfield of Q(ξ|H|) of index [G : H]/2 = 2 . Since n − k = 0 or 2k−1 i n − k = 1, ξ2n−k+1 ∈ F (ξ4). Therefore, F (ξ4) = Q(ξ|H| ) = Q(ξ2n−k+1 ). We take xi = ξ|H| 2n−2+i i k−1 −1 and x2k−1+i = ξ|H| = ξ4ξ|H|, for i = 0,..., 2 − 1. Hence, if i 6= j, then xixj is either ±1 ±i ±1 ±i k−1 i ξ4 or ξ|H| or ξ4 ξ|H|, with i = 1,..., 2 − 1. Since, ξ4 ∈ F (ξ4) and ξ|H| ∈/ F (ξ4) for k−1 ±i i = 1,..., 2 − 1, we have trQ(ξ|H|)/F (ξ4)(ξ|H|) = 0. Therefore

±1 ±1 ±1 trQ(ξ|H|)/F (ξ4 ) = trF (ξ4)/F (trQ(ξ|H|)/F (ξ4)(ξ4 )) = [Q(ξ|H|): F (ξ4)]trF (ξ4)/F (ξ4 ) = 0,

±i ±i trQ(ξ|H|)/F (ξ|H|) = trF (ξ4)/F (trQ(ξ|H|)/F (ξ4)(ξ|H|)) = 0, and

±1 ±i ±1 ±i ±1 ±i trQ(ξ|H|)/F (ξ4 ξ|H|) = trF (ξ4)/F (trQ(ξ|H|)/F (ξ4)(ξ4 ξ|H|)) = trF (ξ4)/F (ξ4 trQ(ξ|H|)/F (ξ4)(ξ|H|)) = 0.

−1 We deduce that tr (x x ) = 0 for every i 6= j. Then the elements x , . . . , x k satisfy Q(ξ|H|)/F i j P 0 2 −1 e 1 the conditions of Lemma 4.1, and hence the conjugates of b = [G:H] gH∈G/H gH by

−1 −1 −1 2 −1 2k−1−1 φ (x0) = 1, φ (x1) = a, φ (x2) = a , . . . , φ (x2k−1−1) = a ,

−1 2n−2 −1 2n−2+1 −1 2n−2+2k−1−1 φ (x2k−1 ) = a , φ (x2k−1+1) = a , . . . , φ (x2k−1) = a

4.1. The rational group ring of nilpotent groups 81 Chapter 4. A complete set of orthogonal primitive idempotents form a complete set of orthogonal primitive idempotents of QGe. In this case, b20 = 1, e T20 = {1}, β2 = b and

2 [G:H]/2−1 2n−2 2n−2+1 2n−2+[G:H]/2−1 T2 = {1, a, a , . . . , a , a , a , . . . , a }.

This is case (1)(ii) and we see that the statement of the Theorem holds. k+1 Now assume that G ' P2. Then [G : H] = 2 and v2(r − 1) = n − k ≥ 2 (see proof of 2k −2k 2k −2k −1 Lemma 4.5), therefore a + a is central and hence, F = Q(ξ|H| + ξ|H| ) = Q(ξ2n−k + ξ2n−k ). k 2k Since ξ4 ∈/ F , F (ξ4) has index 2 in Q(ξ|H|), hence F (ξ4) = Q(ξ|H|) = Q(ξ2n−k ). We take i 2n−2+i i k xi = ξ|H| and x2k+i = ξ|H| = ξ4ξ|H|, for i = 0,..., 2 − 1. By the same argument as in the previous case , the elements x , . . . , x k+1 satisfy the conditions of Lemma 4.1, and hence 0 P 2 −1 ß 1 the conjugates of hb, ci = [G:H] gH∈G/H gH by

−1 −1 −1 2 −1 2k−1 φ (x0) = 1, φ (x1) = a, φ (x2) = a , . . . , φ (x2k−1) = a ,

−1 2n−2 −1 2n−2+1 −1 2n−2+2k−1 φ (x2k ) = a , φ (x2k+1) = a , . . . , φ (x2k+1−1) = a form a complete set of orthogonal primitive idempotents of QGe. In this case, b20 = 1, ß T20 = {1}, β2 = hb, ci and

2 [G:H]/2−1 2n−2 2n−2+1 2n−2+[G:H]/2−1 T2 = {1, a, a , . . . , a , a , a , . . . , a }.

This is case (1)(ii) and we see that the statement of the Theorem holds. Now assume that G ' P3, i.e. H does not have a complement in G. Then F has index k+1 2k −2k [G : H] = 2 in Q(ξ|H|) and v2(r − 1) = n − k ≥ 2 or k = 0. Therefore a + a is 2k −2k −1 e e e e central and hence, F = Q(ξ|H| + ξ|H| ) = Q(ξ2n−k + ξ2n−k ). Moreover, bQGb = bQha, cib. Consider the classical crossed product A = Q(ξ|H|) ∗ hbHi ⊂ Q(ξ|H|) ∗ G/H. The twisting 2k in A is trivial since hbi ∩ H = {1} and the center is Q(ξ|H|). Following the same arguments e 2k−1 as in the case G ' P1, we obtain that the conjugates of b by the elements 1, a, . . . , a e e k are orthogonal idempotents. Therefore, we have that bajb = 0 for all aj ∈/ Q(a2 ). Also k n−2 Q(a2 ) = φ−1(F ) + φ−1(F )a2 . Therefore,

e e e 2k e −1 −1 2n−2 −1 −1 2n−2 bQha, cib = bQha , cib ' φ (F ) + φ (F )a + φ (F )c + φ (F )a c ' H(F ). e Since F is a real ﬁeld, H(F ) is a division ring. Therefore b is the only nonzero idempo- e e e e e 0 tent of bQGb and hence b is a primitive idempotent of QGe. Note also that btQGbt ' e e et et0 2 2k−1 bQGb and M2k (H(F )) ' ⊕t,t0∈T b QGb ⊆ QG, for T = {1, a, a , . . . , a }. However, et et0 2k+2 2 dimF (⊕t,t0∈T b QGb ) = 2 = [G : H] = dimF (QG), hence QGe ' M2k (H(F )). So, one can provide the 2k orthogonal primitive idempotents needed in this case by taking the e 2 2k−1 e conjugates of b by 1, a, a , . . . , a . In this case, b20 = 1, T20 = {1}, m = 1, β2 = b and 2 2k−1 T2 = {1, a, a , . . . , a }. This is case (2)(i) and we see that the statement of the Theorem holds. This ﬁnishes the p-group case. Let us now consider the general case where G is not necessarily a p-group. Since G is

0 ﬁnite nilpotent, G = G2 × Gp1 × · · · × Gpr = G2 × G2 , with pi an odd prime for every i = 1, . . . , r. Then (H, 1) is a strong Shoda pair of G if and only if (Hpi , 1) is a strong

Shoda pairQ of Gpi , for every i = 0, 1, . . . , r, with p0 = 2, and (H) = (H2 × Hp1 × · · · × r Hpr ) = i=0 (Hpi ). It easily follows using the universal property of tensor products that

4.1. The rational group ring of nilpotent groups 82 Chapter 4. A complete set of orthogonal primitive idempotents

Q r r QG(H) = Q( i=0 Gpi (Hpi )) ' ⊗i=0QGpi (Hpi ), the tensor product over Q of the simple algebras QGpi (Hpi ). On the other hand, we have seen that for i ≥ 1, G ' P1 and that ni ki QGpi (Hpi ) ' (Q(ξ|Hp |)/Q(ξ ni−ki ), 1) ' M ki (Q(ξ ni−ki )), for pi = |Hpi |, pi = [Gpi : Hpi ]. i pi pi pi Since, in general,

Mn( (ξs)) ⊗ Mm( (ξt)) ' Mnm( (ξlcm(s,t))) Q Q Q 2 Q 3 a11B a12B ··· a1nB 6 7 6 7 6 a21B a22B ··· a2nB 7 A ⊗ B 7→ 6 . . . . 7 Q 4 . . .. . 5 an1B an2B ··· annB is an isomorphism, we have that

G 0 (H 0 ) ' ⊗ G (H ) ' ⊗ M k ( (ξ n −k )) ' M ( (ξ )), Q 2 2 i≥1Q pi pi i≥1 i Q i i [G20 :H20 ] Q m pi pi with m = |H20 |/[G20 : H20 ] = [H20 : K]/[G20 : H20 ] and then a complete set of orthogonal primitive idempotents of QG20 (H20 ) can be obtained by multiplying the diﬀerent sets of idempotents obtained for each tensor factor. Since each Gpi , with i ≥ 1 takes the form haii o hbii and so, G20 = (ha1i × · · · × hari) o (hb1i × · · · × hbri) = hai o hbi, with a = a1 ··· ar and b = b1 ··· br, since o(ai) and o(aj) (respectively o(bi) and o(bj)) are coprime. Remember pki 0 that ai is central in Gpi and hence also in G2 . The product of the diﬀerent primitive Ü idempotents of the factors from the odd part is the product of the conjugates of bi(Hpi ) by ki 2 pi −1 1, ai, ai , . . . , ai . This can be written as the set

i i i 1 2 r kj Ü a1 Ü a2 Ü ar {(b1(Hp1 )) (b2(Hp2 )) ··· (br(Hpr )) | ij = 0, 1, . . . , pj − 1}.

Since the aj’s commute with everything except with bj, this set is equal to the set

i i i 1 2 r kj Ü Ü Ü a1 a2 ···ar {(b1b2 ··· br(H20 )) | ij = 0, 1, . . . , pj − 1}. Ü Ü Ü e Since all hbii are disjunct, b1b2 ··· br = b. Using the Chinese Remainder Theorem, we obtain nj for every set {i1, . . . , ir} an integer x such that x ≡ ij mod pj for all j = 1, . . . , r. Then i1 i2 ir x a1 a2 ··· ar = a . Moreover, a complete set of orthogonal primitive idempotents of QG20 (H20 ) is of the form e ax {(b(H20 )) | x = 0, 1,..., [G20 : H20 ] − 1},

k1 k2 kr [G 0 :H 0 ] p p ···pr since a 2 2 = a 1 2 is central. In the notation of the Theorem, we have that a20 = a, 2 [G 0 :H 0 ]−1 b20 = b and T20 = {1, a, a , . . . , a 2 2 }, as wanted. If |G| is odd, then β2 = 1 and T2 = {1} (case (1)(i)) and the proof is ﬁnished. Otherwise we should combine the odd and even parts of G. If H2 has a complement in

G2, then QG2(H2) ' (Q(ξ|H2|)/F2, 1) ' M[G2:H2](F2) and we say that QG2(H2) is split over its center. In this case we can take T2 and β2 as in the 2-group case. Then, a complete set of Ý t 0 t2 orthogonal primitive idempotents of QG consists of the elements (b20 (H20 )) 2 (β2(H2)) = Ý t 0 t2 (b20 β2) 2 , with t20 ∈ T20 and t2 ∈ T2, as wanted (case (1)(i) or (1)(ii)). −1 If H2 does not have a complement in G2, then QG2(H2) ' M[G2:H2]/2(H(Q(ξ2n−k +ξ2n−k ))), n k with |H2| = 2 and [G2 : H2]/2 = 2 . Hence −1 Ge ' M ( ( (ξ n−k + ξ ))) ⊗ M ( (ξ )) Q [G2:H2]/2 H Q 2 2n−k Q [G20 :H20 ] Q m −1 ' M[G:H]/2(H(Q(ξm, ξ2n−k + ξ2n−k ))),

4.1. The rational group ring of nilpotent groups 83 Chapter 4. A complete set of orthogonal primitive idempotents

−1 with m = |H20 |/[G20 : H20 ]. If H(Q(ξm, ξ2n−k + ξ2n−k )) is not split, or equivalently the conditions of (2)(i) hold, then we can also take T2 and β2 as in the 2-group case. However, if −1 −1 −1 H(Q(ξm, ξ2n−k + ξ2n−k )) is split, then H(Q(ξm, ξ2n−k + ξ2n−k )) ' M2(Q(ξm, ξ2n−k + ξ2n−k )) and [G20 :H20 ] [G20 :H20 ] one should duplicate the number of idempotents. Remember that φ(a 0 ) = ξ = ξm 2 |H20 | k k k k 2 2 [G20 :H20 ] 2 −2 and φ(a ) = ξ = ξ n−k . We deduce that there exist x, y ∈ (a 0 , a +a ) such that 2 |H2| 2 Q 2 2 2 2 2 1 2n−2 2n−2 1 2n−2 2n−2 (1+x +y ) = 0 and f = 2 (1+xa2 +ya2 c2) and 1−f = 2 (1−xa2 −ya2 c2) form a −1 complete set of primitive orthogonal idempotents of H(Q(ξm, ξ2n−k + ξ2n−k )) (see Lemma 4.6). Ü 2 2k−1 Recall that the idempotents of QG2(H2) are the conjugates of b2(H2) by 1, a2, a2, . . . , a2 , Ý and that the idempotents of QG20 (H20 ) are the conjugates of b20 (H20 ) by the elements of 2 [G20 :H20 ]−1 c2 Ü T20 = {1, a20 , a20 , . . . , a20 }. Observe that 1−f = f . Now deﬁne β2 = b2f and consider Ü0 2 2k−1 2 2k−1 the conjugates of b2β2 by T20 T2, with T2 = {1, a2, a2, . . . , a2 , c2, c2a2, c2a2, . . . , c2a2 }. We will prove that these elements are orthogonal. Hence we will have found a complete set of orthogonal primitive idempotents of QGe. Note that

i1 j1 k1 i2 j2 k2 ÝÜ a 0 c a ÝÜ a 0 c a α := (b20 b2f) 2 2 2 (b20 b2f) 2 2 2 i1 i2 a j1 k1 a j2 k2 Ý 20 Ü c a Ý 20 Ü c a = b20 (b2f) 2 2 b20 (b2f) 2 2 i1 i2 k1 k2 a a a a j1 k1 j2 k2 Ý 20 Ý 20 Ü 2 Ü 2 c a c a = b20 b20 b2 b2 f 2 2 f 2 2 .

i1 j1 k1 i2 j2 k2 Assume a20 c2 a2 6= a20 c2 a2 . If i1 6= i2 or k1 6= k2 then α = 0. Otherwise, j1 6= j2 and we have cj1 ak1 cj2 ak1 ak1 ak1 that α = γf 2 2 f 2 2 = γf 2 (1 − f) 2 = 0 for some γ ∈ QG. Hence, these idempotents are orthogonal. This proves the statement in case (2)(ii). 2

Remark 4.8. We can also create another complete set of orthogonal primitive idempotents of QGe in case (2)(ii). In the proof of this case, we saw that we have to duplicate the number of idempotents. Another way to do this, is to multiply the obtained orthogonal 2n−2 2n−2 1+xa2 +ya2 c2 idempotents of QGe with the orthogonal primitive idempotents f = 2 and 2n−2 2n−2 1−xa2 −ya2 c2 −1 1 − f = 2 of H(Q(ξm, ξ2n−k + ξ2n−k )). This will result in the complete set

Ý t 0 t2t Ý t 0 t2t {(b20 β2) 2 f, (b20 β2) 2 (1 − f) | t20 ∈ T20 , t2 ∈ T2, t ∈ TG/N }

2 2k−1 of orthogonal primitive idempotents of QGe, with T2 = {1, a2, a2, . . . , a2 } and T20 = 2 [G20 :H20 ]−1 {1, a20 , a20 , . . . , a20 }. Let us now simplify the notation of Theorem 4.7.

Notation 4.9. Let G be a ﬁnite nilpotent group, (H,K) a strong Shoda pair of G and e = e(G, H, K). A complete set of orthogonal primitive idempotents of QGe consists of the Ý conjugates βe = b20 β2 by the elements of Te = T20 T2TG/N , where = (H,K), TG/N denotes [N20 :H20 ]−1 a right transversal of N = NG(K) in G, T20 = {1, a20 , . . . , a20 }, 8 > 2k−1 < {1, a2, . . . , a2 }, in cases (1)(i) and (2)(i) n−2 [N2:H2]/2−1 2n−2 2n−2+1 2 +[N2:H2]/2−1 T2 = > {1, a2, . . . , a2 , a2 , a2 , . . . , a2 }, in cases (1)(ii) : 2k−1 2k−1 {1, a2, . . . , a2 , c2, c2a2, . . . , c2a2 }, in cases (2)(ii)

4.1. The rational group ring of nilpotent groups 84 Chapter 4. A complete set of orthogonal primitive idempotents and 8 g <> M2 in case (1) Ü β2 = > b2 in case (2)(i) : 2n−2 2n−2 Ü 1+xa2 +ya2 c2 b2 2 in case (2)(ii).

Remark 4.10. A description of the simple algebras QGe using Theorem 2.17 can be given according to the cases listed above. Thus, QGe ' M[G:N](M[N:H](F )) ' M[G:H](F ), with F the fixed field of the natural action of N/H ' Gal(Q(ξ[H:K]/F ) on the cyclotomic field Q(ξ[H:K]) = QH(H,K), in cases (1) and (2)(ii) of Theorem 4.7, and QGe ' M[G:N](M[N:H]/2(H(F ))) = −1 M[G:N]/2(H(F )), in case (2)(i) and in this case F = Q(ξm, ξ2n +ξ2n ), with m = [H20 : K]/[N20 : n H20 ] and 2 = 2[H2 : K]/[N2 : H2]. Using this and Lemma 4.6, one deduces that if QGe is a non-commutative division ring, then only case (2)(i) can occur with [G : H] = 2, G = N (since H2 6= N2) and m = 1 or n ≤ 2 and the multiplicative order of 2 modulo m is odd. If m = 1, −1 n we have that QGe ' H(Q(ξ2n + ξ2n )) with 2 = [H : K]. Otherwise, QGe ' H(Q(ξm)), with m odd such that the order of 2 modulo m is odd and [H : K] = 2nm = m or 2m or 4m.

Notice that the use of Lemma 4.1 has been essential in all the cases of the proof of Theorem 4.7. We would like to be able to give a similar description to the one from Theorem 4.7 for a complete set of orthogonal primitive idempotents for rational group algebras of arbitrary finite metacyclic groups. Unfortunately, the approach of Theorem 4.7 does not apply here. For −1 2 example, if G = C7 o C3 = hai o hbi, with b ab = a and e = e(G, hai, 1) = (hai), then there is not a complete set of orthogonal primitive idempotents of QGe formed by Q(a)-conjugates e √ of b. This is a consequence of Example 4.3. Indeed, observe that QGe ' (Q(ξ7)/Q( −7), 1), byP Theorem 2.17. Then there does not exist x1, x2, x2 ∈ Q(ξ7) such that the conjugates of 1 √ g by x , x , x form a complete set of orthogonal primitive idempotents of 3 g∈Gal(Q(√ξ7)/Q( −7)) 1 2 3 √ (Q(ξ7)/Q( −7), 1). Note that hbi ' G/hai ' Gal(Q(ξ7)/Q( −7)). So, if φ : Q(a) → Q(ξ7) is the isomorphism defined by a 7→ ξ7, then there does not exist elements in Q(a) such that e the conjugates of b by these elements of Q(a) form a complete set of orthogonal primitive idempotents of QGe. Next we will describe a complete set of matrix units in a simple component QGe, where e = e(G, H, K) with (H,K) a strong Shoda pair of a finite nilpotent group G. We will use Notation 4.9.

Corollary 4.11. Let G be a ﬁnite nilpotent group. For every primitive central idempotent e = e(G, H, K), with (H,K) a strong Shoda pair of G, let Te and βe be as in Notation 4.9 0 and set N = NG(K). For every t, t ∈ Te, let

−1 0 Ett0 = t βet . P 0 0 Then {Ett | t, t ∈ Te} gives a complete set of matrix units in QGe, i.e. e = t∈Te Ett and

Et1t2 Et3t4 = δt2t3 Et1t4 , for every t1, t2, t3, t4 ∈ Te. Moreover EttQGEtt ' F , in case (1) and (2)(ii) of Theorem 4.7, and EttQGEtt ' H(F ), in case (2)(i) of Theorem 4.7, where F is the ﬁxed subﬁeld of QH(H,K) under the natural action of N/H.

Proof. We know from Theorem 4.7 that the set {Ett | t ∈ Te} is a completeP set of orthogonal primitive idempotents of QGe. Therefore it easily follows that e = t∈Te Ett and Et1t2 Et3t4 =

4.1. The rational group ring of nilpotent groups 85 Chapter 4. A complete set of orthogonal primitive idempotents

δt2t3 Et1t4 , for t1, t2, t3, t4 ∈ Te. Let F be the ﬁxed subﬁeld of QH under the natural action of N/H. From Remark 4.10, we know that QGe = Mn(F ) in cases (1) and (2)(ii) of Theorem 4.7, and that QGe = Mn(H(F )) in case (2)(i), with n = |Te|. A computation shows that

QGe ' EndQG(QGe) = EndQG(⊕t∈Te QGEtt) n ' EndQG((QGEtt) ) ' Mn(EndQG(QGEtt)) ' Mn(EttQGEtt).

Hence EttQGEtt is isomorphic to F or to H(F ). 2

4.2 Examples

Theorem 4.7 gives an algorithmic approach to describe a complete set of orthogonal primitive idempotents. Hence, we can implement it in GAP. This is done in the GAP function OrthogonalPrimitiveIdempotents, which can be found in Chapter5. In this section, we will use this function to compute complete sets of orthogonal primitive idempotents in several examples. This section is based on own research.

Example 4.12. In Example 2.38, we computed the Wedderburn decomposition of Q8 and discovered that QQ8 ' Q ⊕ Q ⊕ Q ⊕ Q ⊕ H(Q). We compute a complete set of orthogonal primitive idempotents of each simple component. gap> set:=OrthogonalPrimitiveIdempotents(QG); [ rec ( Idempotents := [ (1/8)*x^4+(1/8)*x^2*y*x^3*y+(1/8)*y+(1/8)*x^2*y*x ^2*y+(1/8)*x^3*y*x^2+(1/8)*x^3*y*x^2*y+(1/8)*y*x^2+(1/8)*x^3*y ] ), rec ( Idempotents := [ (1/8)*y*x^-3*y*x^-1+(-1/8)*x^-2*y*x^-2*y*x^-1+(-1 /8)*x^-5*y*x^-1+(1/8)*x^-2*y*x^-3*y*x^-1+(1/8)*x^-2*y*x^ -3+(-1/8)*x^-2*y*x^-4*y*x^-1+(-1/8)*x^-3*y*x^-1+(1/8)*y*x^ -3 ] ), rec( Idempotents := [ (1/8)*x^-2*y*x^-4*y+(-1/8)*x^-2*y*x^-1*y+(1/ 8)*x^-4*y+(1/8)*x^-2*y*x^-2*y+(-1/8)*x^-3*y+(-1/8)*y*x^ -1*y+(1/8)*x^-6*y+(-1/8)*x^-1*y ] ), rec( Idempotents := [ (1/8)*x^3*y^-2*x^3+(1/8)*x^2*y^-2*x+(-1/8)*x*y^ -1*x^3+(1/8)*x^3*y^-2*x+(-1/8)*x^2*y^-1*x^3+(1/8)*x^2*y^ -2*x^3+(-1/8)*x^3*y^-1*x^3+(-1/8)*y^-1*x^3 ] ), rec( Idempotents := [ (1/2)*x^4+(-1/2)*x^2 ] ) ] gap> rc:=StrongShodaPairsAndIdempotents(QG);; gap> for i in [1..Size(set)] do; > e:=CentralElementBySubgroups(QG,rc.("StrongShodaPairs")[i][1], > rc.("StrongShodaPairs")[i][2]); > Print(e=set[i].("Idempotents")[1]); > od; true true true true true

4.2. Examples 86 Chapter 4. A complete set of orthogonal primitive idempotents

We see that for each simple component, a complete set of orthogonal primitive idempotents consists of only one element, namely the primitive central idempotent associated to this component. This is correct, since each simple factor is a division ring.

Example 4.13. In Example 2.39, we computed the Wedderburn decomposition of D8 and saw that QD8 ' Q ⊕ Q ⊕ Q ⊕ Q ⊕ M2(Q). We compute a complete set of orthogonal primitive idempotents for each simple component. gap> set:=OrthogonalPrimitiveIdempotents(QG); [ rec ( Idempotents := [ (1/8)*a^4+(1/8)*b*a^3*b+(1/8)*a^4*b+(1/8)*a^2*b^2 +(1/8)*a*b+(1/8)*a^3+(1/8)*b*a^2+(1/8)*a^3*b ] ), rec( Idempotents := [ (1/8)*a^-2*b*a^-3*b*a^-1+(-1/8)*a^-2*b*a^-4*b*a^-1+(-1/8)*a^ -5*b*a^-1+(1/8)*b*a^-3*b*a^-1+(1/8)*a^-2*b*a^-3+(-1/8)*a^ -2*b*a^-2*b*a^-1+(-1/8)*a^-3*b*a^-1+(1/8)*b*a^-3 ] ), rec( Idempotents := [ (1/8)*a^3*b^-2*a+(1/8)*a^2*b^-2*a^3+(-1/8)*a*b^-1*a+(1/8)*a^ 3*b^-2*a^3+(-1/8)*a^2*b^-1*a+(1/8)*a^2*b^-2*a+(-1/8)*a^3*b^ -1*a+(-1/8)*b^-1*a ] ), rec( Idempotents := [ (1/8)*a^-2*b*a^-2*b+(-1/8)*b*a^-1*b+(1/8)*a^ -4*b+(1/8)*a^-2*b*a^-4*b+(-1/8)*a^-3*b+(-1/8)*a^-2*b*a^ -1*b+(1/8)*a^-6*b+(-1/8)*a^-1*b ] ), rec( Idempotents := [ (1/4)*b*a^-4*b+(-1/4)*b*a^-2*b+(1/4)*b*a^-5+(-1/ 4)*b*a^-3, (1/4)*a^-1*b*a^-4*b*a+(-1/4)*a^-1*b*a^-2*b*a+(-1/ 4)*a^-1*b*a^-2+(1/4)*a^-1*b*a^-4 ] ) ] gap> rc:=StrongShodaPairsAndIdempotents(QG);; gap> set[1].("Idempotents")[1]=rc.("PrimitiveCentralIdempotents")[1]; true gap> set[2].("Idempotents")[1]=rc.("PrimitiveCentralIdempotents")[2]; true gap> set[3].("Idempotents")[1]=rc.("PrimitiveCentralIdempotents")[3]; true gap> set[4].("Idempotents")[1]=rc.("PrimitiveCentralIdempotents")[4]; true gap> e:=rc.("PrimitiveCentralIdempotents")[5]; (1/2)*a^-4+(-1/2)*a^-2 gap> e1:=set[5].("Idempotents")[1]; (1/4)*b*a^-4*b+(-1/4)*b*a^-2*b+(1/4)*b*a^-5+(-1/4)*b*a^-3 gap> e2:=set[5].("Idempotents")[2]; (1/4)*a^-1*b*a^-4*b*a+(-1/4)*a^-1*b*a^-2*b*a+(-1/4)*a^-1*b*a^-2+(1/4)*a^ -1*b*a^-4 gap> e=e1+e2; true gap > e1*e2; of ...

For the 4 simple components isomorphic to Q, we have only one primitive idempotent equal to the primitive central idempotent. For the component, M2(Q), we have a set of 2 orthogonal primitive idempotents which sum up to the primitive central idempotent associated to that 1 2 factor, more precisely e = e1 + e2, with e = 2 (1 − a ) and 1 e = (1 − a2 + ab − a3b) 1 4

4.2. Examples 87 Chapter 4. A complete set of orthogonal primitive idempotents and 1 e = (1 − a2 − ab + a3b). 2 4 However, these examples are nice, we want some examples where we have to combine odd and even parts. This is the case in the following Example.

Example 4.14. Consider the group G = D8 × C3. This group is nilpotent, hence we can compute a complete set of orthogonal primitive idempotents of QG. gap> F:=FreeGroup("a","b","c");;a:=F.1;;b:=F.2;;c:=F.3;; gap> G:=F/[a^4,b^2,b*a*b*a,c^3,a^(-1)*c^(-1)*a*c,b^(-1)*c^(-1)*b*c];; gap> QG:=GroupRing(Rationals,G);; gap> SimpleFactorsFromSSP(G); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, CF(3) ], [ 2, Rationals ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 2, CF (3) ] ] gap> rc:=StrongShodaPairsAndIdempotents(QG);; gap> set:=OrthogonalPrimitiveIdempotents(QG);; gap> e:=rc.("PrimitiveCentralIdempotents")[6]; (1/6)*a^-4+(1/6)*a^-4*c^2+(-1/6)*a^-2+(1/6)*a^-4*c+(-1/6)*a^-2*c^2+(-1/ 6)* a^ -2*c gap> e1:=set[6].("Idempotents")[1]; (1/12)*b*c*a^-4*b*c^2+(1/12)*b*a^-4+(1/12)*b*c^2*a^-4*b+(-1/12)*b*c^2*a^ -2*b*c+(1/12)*c^2*a^-4*b+(-1/12)*b*c*a^-2*c^2+(1/12)*c^2*a^-4*c^2+(-1/ 12)*c^2*a^-2+(1/12)*c^2*a^-4*b*c^2+(-1/12)*c^2*a^-2*b+(-1/12)*c*a^-2+( -1/12)*b*c*a^-2 gap> e2:=set[6].("Idempotents")[2]; (1/12)*a^-1*b*c*a^-4*b*c^2*a+(-1/12)*a^-1*b*c*a^-2*c^2*a+(1/12)*a^ -1*b*c^2*a^-4*b*a+(-1/12)*a^-1*b*c^2*a^-2*b*c*a+(-1/12)*a^-1*c^2*a^ -2*b*a+(1/12)*a^-1*b*a^-3+(1/12)*a^-1*c^2*a^-4*c^2*a+(-1/12)*a^-1*c^2*a^ -1+(-1/12)*a^-1*b*c*a^-1+(1/12)*a^-1*c^2*a^-4*b*a+(-1/12)*a^-1*c*a^-1+( 1/12)*a^-1*c^2*a^-4*b*c^2*a gap> e=e1+e2; true gap > e1*e2; of ... gap> f:=rc.("PrimitiveCentralIdempotents")[10]; (1/3)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1+(-1/6)*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1+(-1/3)*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ 2*b+(-1/6)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1+(1/6)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^2*b+(1/6)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^2*b gap> f1:=set[10].("Idempotents")[1]; (1/6)*b*a*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^2*b+(-1/12)*b*a*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^2*b+(-1/ 6)*b*a*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1+(1/6)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^2*b+(-1/ 12)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1+(1/12)*b*a*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1+(-1/12)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^

4.2. Examples 88 Chapter 4. A complete set of orthogonal primitive idempotents

-1*c*b*a^-1*c*b*a^-1*c*b*a^2*b+(-1/6)*b*a*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^2*b+(1/ 12)*b*a*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1+( -1/12)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^2*b+(1/12)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1+(1/12)*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1 gap> f2:=set[10].("Idempotents")[2]; (1/6)*a^-1*b*a*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^2*b*a+(-1/12)*a^ -1*b*a*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^2*b*a+(-1/6)*a^-1*b*a*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b+(-1/6)*a^-1*b*a*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ 2*b*a+(-1/12)*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b+(1/12)*a^-1*b*a*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b+(1/12)*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b+(1/6)*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^2*b*a+(1/12)*a^ -1*b*a*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b+(1/ 12)*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b+(-1/ 12)*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^2*b*a+(-1/12)*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^ -1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^-1*c*b*a^2*b*a gap> f=f1+f2; true gap > f1*f2; of ...

We see that QG ' 4Q ⊕ 4Q(ξ3) ⊕ M2(Q) ⊕ M2(Q(ξ3)), and that we have one idempotent for each factor, except for the factors M2(Q) and M2(Q(ξ3)). For these factors, we have 2 orthogonal primitive idempotents.

Example 4.15. Consider the group G = (ha | a3 = 1i × hb | b3 = 1i) o hc | c3 = 1i, with c acting on a and b as ac = a and bc = ba. This group is nilpotent and we can compute a complete set of orthogonal primitive idempotents of QG. gap> F:=FreeGroup("a","b","c");;a:=F.1;;b:=F.2;;c:=F.3;; gap> G:=F/[a^3,b^3,c^3,a^(-1)*b^(-1)*a*b,a^(-1)*c^(-1)*a*c, > c^(-1)*b*c*a^(-1)*b^(-1)];; gap> QG:=GroupRing(Rationals,G);; gap> SimpleFactorsFromSSP(G); [ [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 3, CF (3) ] ] gap> rc:=StrongShodaPairsAndIdempotents(QG);; gap> set:=OrthogonalPrimitiveIdempotents(QG);; gap> e:=rc.("PrimitiveCentralIdempotents")[6]; (2/3)*c*b^-1*c*b^-1*c*b^-1+(-1/3)*c*b^-1*c*b*c+(-1/3)*c*b*c*b^2*c gap> e1:=set[6].("Idempotents")[1]; (2/9)*c^-1*b*c*b^-1*c*b^-1*c*b*c+(-1/9)*c^-1*b*c^-1*b*c*b^-1*c*b^-1+(-1/ 9)*c^-1*b*c*b^-1+(2/9)*c^-1*b*c^-1*b*c*b^-1*c*b^-1*c*b^-1+(-1/9)*c^ -1*b*c*b^-1*c*b*c+(-1/9)*c*b^-1*c*b*c*b^2*c+(-1/9)*c*b*c+(-1/9)*c^ -1*b*c^-1*b*c*b*c*b^2*c+(2/9)*c^-1*b*c*b^-1*c*b^-1*c*b^-1

4.2. Examples 89 Chapter 4. A complete set of orthogonal primitive idempotents gap> e2:=set[6].("Idempotents")[2]; (2/9)*b^-1*c^-1*b*c*b^-1*c*b^-1*c*b*c*b+(-1/9)*b^-1*c^-1*b*c^-1*b*c*b^ -1*c+(-1/9)*b^-1*c^-1*b*c+(-1/9)*b^-1*c^-1*b*c^-1*b*c*b*c*b^2*c*b+(-1/ 9)*b^-1*c*b*c*b+(2/9)*b^-1*c^-1*b*c^-1*b*c*b^-1*c*b^-1*c+(2/9)*b^-1*c^ -1*b*c*b^-1*c*b^-1*c+(-1/9)*b^-1*c*b^-1*c*b*c*b^2*c*b+(-1/9)*b^-1*c^ -1*b*c*b^-1*c*b*c*b gap> e3:=set[6].("Idempotents")[3]; (2/9)*b*c^-1*b*c*b^-1*c*b^-1*c*b*c*b^-1+(-1/9)*b*c^-1*b*c^-1*b*c*b^ -1*c*b^-2+(-1/9)*b*c^-1*b*c*b^-2+(-1/9)*b*c*b^-1*c*b*c*b^2*c*b^-1+(2/ 9)*b*c^-1*b*c*b^-1*c*b^-1*c*b^-2+(-1/9)*b*c^-1*b*c^-1*b*c*b*c*b^2*c*b^ -1+(-1/9)*b*c^-1*b*c*b^-1*c*b*c*b^-1+(2/9)*b*c^-1*b*c^-1*b*c*b^-1*c*b^ -1*c*b^-2+(-1/9)*b*c*b*c*b^-1 gap> e=e1+e2+e3; true gap > e1*e2; of ... gap > e2*e3; of ... gap > e1*e3; of ... We see that QG ' Q ⊕ 4Q(ξ3) ⊕ M3(Q(ξ3)), and that we have one idempotent for each factor, except for the factor M3(Q(ξ3)). For this factor, we have 3 orthogonal primitive idempotents.

Our algorithm seems to work very well. However up to now, we have not dealt with case (2)(ii) of Theorem 4.7. As far as we know, GAP can not compute roots of the equation k k 2 2 [N20 :H20 ] 2 −2 (1 + x + y ) = 0 in Q(a20 , a2 + a2 ). Therefore, our GAP function will compute a 2n−2 2n−2 1+xa2 +ya2 c2 complete set of orthogonal primitive idempotents up to multiplication with 2 2n−2 2n−2 1−xa2 −ya2 c2 and 2 . By Remark 4.8, we know that this is possible. However, this shows the limitations of the method. We will give an example.

Example 4.16. Consider the group ha, b | a4 = b12 = 1, b−1ab = a−1i. gap> F:=FreeGroup("a","b");;a:=F.1;;b:=F.2;; gap> G:=F/[a^4,b^12,b^(-1)*a*b*a];; gap> a:=G.1;;b:=G.2;; gap> QG:=GroupRing(Rationals,G);; gap> H:=StrongShodaPairs(G)[16][1]; Group([ a, b^-2 ]) gap> K:=StrongShodaPairs(G)[16][2]; Group([ a^2*b^6 ]) gap> Size(H); 24 gap> Size(K); 2 gap> SimpleAlgebraByStrongSPInfo(QG,H,K); [ 1, CF(3), 12, [ [ 2, 7, 10 ] ], [ ] ] gap> A:=SimpleAlgebraByStrongSP(QG,H,K);; gap> IsDivisionRing(A); false gap> set:=CompleteSetOrthogonalPrimitiveIdempotents(QG,H,K);

4.2. Examples 90 Chapter 4. A complete set of orthogonal primitive idempotents

Warning: This set is not complete! One should multiply these idempotents with 1/2(1+x*a2^power+y*a2^power*c2) and 1/2(1-x*a2^power-y*a2^power*c2 ), where x and y are solutions of (1+x^2+y^2)epsilon=0 in Q(v,w+w^-1)eps ilon rec( Idempotents := [ (1/6)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2+(-1/12)*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2+(-1/6)*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6+(-1/6)*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2+(-1/12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2+(1/12)*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6+(1/ 12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2+(1/6)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6+(1/12)*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6+(1/12)*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2+(-1/12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^2*b^6+(-1/12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6 ], v := b^-4, w := a, a2 := a, power := 1, c2 := b^3, epsilon := (1/6)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2+(-1/12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2+(-1/6)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6+(-1/ 6)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2+(-1/12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2+(1/12)*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6+(1/12)*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2+(1/6)*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^

4.2. Examples 91 Chapter 4. A complete set of orthogonal primitive idempotents

2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^2*b^6+(1/12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6+(1/ 12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2+(-1/12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^2*b^6+(-1/12)*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^ 2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^ -1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^-1*b^2*a^2*b^6 ) gap> homHK:=NaturalHomomorphismByNormalSubgroup(H,K);; gap> hK:=Representative(PreImages(homHK, > MinimalGeneratingSet(FactorGroup(H,K))[1])); a^ -1*b^2 gap> Image(homHK,b^(-4))=Image(homHK,hK^4); true gap> Image(homHK,a)=Image(homHK,hK^9); true

The group G has a strong Shoda pair (H,K), with H = ha, b−2i and K = ha2b6i and H/K = ha−1b2Ki. The simple algebra QGe(G, H, K) has the description

7 2 10 Q(ξ12)(g | ξ12g = gξ12 = −gξ12, g = ξ12 = −ξ3). This is not a division ring, however we have found only one idempotent. But as the warning in the output tells us, we have to multiply this idempotent with 1/2(1 + xa + yab3) and 1/2(1 − xa − yab3), where x and y are solutions of (1 + x2 + y2)(H,K) = 0 in Q(b−4, a + a−1)(H,K). −4 −1 2 4 Recall that QH(H,K) ' Q(ξ[H:K]) = Q(ξ12) and we see that b K = (a b ) K and aK = −1 2 9 −4 −1 2 −2 (a b ) K, therefore Q(b , a + a )(H,K) ' Q(ξ3, 0) = Q(ξ3). Since 1 + ξ3 + ξ3 = 0, we have found a solution x = b−4 and y = b4. Now we can compute a complete set of orthogonal idempotents and check if they are indeed orthogonal idempotents. gap> eps:=set.("epsilon");; gap> f:=1/2*(eps+b^(-4)*a*eps+b^4*a*b^3*eps);; gap> f2:=1/2*(eps-b^(-4)*a*eps-b^4*a*b^3*eps);; gap> e1:=set.("Idempotents")[1]*f;; gap> e2:=set.("Idempotents")[1]*f2;; gap> e1=e1*e1; true gap> e2=e2*e2; true gap > e1*e2; of ... gap> e1+e2=CentralElementBySubgroups(QG,H,K); true

In this Example we were able to compute roots of the equation, but in general this is not possible and it is wishful that GAP could solve this. So it would be of interest to many to do some research on algorithms to solve this kind of equations.

4.2. Examples 92 Chapter 4. A complete set of orthogonal primitive idempotents

4.3 Finite group rings of nilpotent groups

The next step is to describe a complete set of orthogonal primitive idempotents of a semisimple finite group algebra FG for a nilpotent group G. In Chapter3, we proved that the primitive central idempotents of a semisimple group algebra FG, over an abelian-by-supersolvable group, are of the form eC (G, H, K), for a strong Shoda pair (H,K) of G and a C ∈ C(H/K). Now we will describe a complete set of primitive orthogonal idempotents of FGeC (G, H, K) for finite nilpotent groups G. This section is based on own research. One can copy the proof of Theorem 4.7 and change things to the finite case. We will do this. First, we have to give a Lemma similar to Lemma 4.1.

Lemma 4.17. Let Fq be a ﬁeld of order q and let A = (Fqn /Fq, 1) be a classical crossed productP with trivial twisting and (n, q) = 1. Let G = Gal(Fqn /Fq) with |G| = n. Let e = −1 |G| g∈G g ∈ A and let x1, . . . , xn be non-zero elements of Fqn . Then the conjugates of e by x1, . . . , xn form a complete set of orthogonal primitive idempotents of A if and only if −1 trFqn /Fq (xixj ) = 0 for every i 6= j. Proof. By Corollary 1.34, the classical crossed product A is isomorphic to Mn(Fq). There- fore, a complete set of orthogonal primitive idempotents of A contains n elements. So we only have to prove that e is an idempotent and that the conjugates of e by x1, . . . , xn are −1 orthogonal if and only if trFqn /Fq (xixj ) = 0 for every i 6= j. Denote tr = tr . As the twisting is trivial, e is an idempotent of A. Moreover, eg = e Fqn /Fq P P −1 −1 Pfor every g ∈ G. Therefore, if x ∈ Fqn , then exe = ex|G| g∈G g = g∈G eg|G| g(x) = −1 −1 ∗ −1 −1 g∈G e|G| g(x) = e|G| tr(x). Thus if xi, xj ∈ Fqn , i 6= j, then xi exi and xj exj are −1 −1 −1 −1 −1 orthogonal if and only if xi exixj e = 0 if and only if xi e|G| tr(xixj ) = 0 if and only if −1 −1 −1 −1 xi |G| tr(xixj ) = 0 if and only if tr(xixj ) = 0. 2

We also need a complete set of orthogonal primitive idempotents of a Hamiltonian quaternion algebra ( ) = −1,−1 . H F F

Lemma 4.18. Let F be a finite field of characteristic not 2. Then the quaternion algebra 2 2 1 H(F) splits if and only if x + y = −1 for some x, y ∈ F. In that case 2 (1 + xi + yj) and 1 2 (1 − xi − yj) form a complete set of orthogonal primitive idempotents of H(F). Proof. The first part is a consequence of Proposition 1.37. If H(F) splits, then it is easy to 1 1 verify that 2 (1+xi+yj) and 2 (1−xi−yj) are orthogonal idempotents of H(F) and their sum is 1. Since in this case H(F) ' M2(F), we have found a complete set of orthogonal primitive idempotents of H(F). 2

Now we state the main Theorem.

Theorem 4.19. Let F be a finite field of order q and G a finite group such that (|G|, q) = 1. Let (H,K) be a strong Shoda pair, C ∈ C(H/K) and set e = eC (G, H, K), = C (H,K), H/K = hai, E = EG(H/K) and let E2/K and H2/K = ha2i (respectively E20 /K and H20 /K = ha20 i) denote the 2-parts (respectively 2’-parts) of E/K and H/K respectively. Then ha20 i has a cyclic complement hb20 i in E20 /K. A complete set of orthogonal primitive idempotents of FGe consists of the conjugates of

4.3. Finite group rings of nilpotent groups 93 Chapter 4. A complete set of orthogonal primitive idempotents

Ý 2 [E20 :H20 ]−1 βe = b20 β2 by the elements of Te = T20 T2TG/E, where T20 = {1, a20 , a20 , . . . , a20 }, TG/E denotes a right transversal of E in G and β2 and T2 are given according to the cases below. g (1) If H2/K has a complement M2/K in E2/K then β2 = M2. Moreover, if M2/K is cyclic, then there exists b2 ∈ E2 such that E2/K is given by the following presentation

n k 2 2 b2 r ha2, b2 | a2 = b2 = 1, a2 = a2 i,

and if M2/K is not cyclic, then there exist b2, c2 ∈ E2 such that E2/K is given by the following presentation

n k 2 2 2 b2 r c2 −1 ha2, b2, c2 | a2 = b2 = 1, c2 = 1, a2 = a2 , a2 = a2 , [b2, c2] = 1i,

2n−2 with r ≡ 1 mod 4 (or equivalently a2 is central in E2/K.) Then

2 2k−1 2n−2 (i) T2 = {1, a2, a2, . . . , a2 }, if a2 is central in E2/K (unless n ≤ 1) and M2/K is cyclic; and n−2 2 [E2:H2]/2−1 2n−2 2n−2+1 2 +[E2:H2]/2−1 (ii) T2 = {1, a2, a2, . . . , a2 , a2 , a2 , . . . , a2 }, otherwise.

(2) If H2/K has no complement in E2/K, then there exist b2, c2 ∈ E2 such that E2/K is given by the following presentation

n k n−1 2 2 2 2 b2 r c2 −1 ha2, b2, c2 | a2 = b2 = 1, c2 = a2 , a2 = a2 , a2 = a2 , [b2, c2] = 1i,

2n−2 2n−2 Ü 1+xa2 +ya2 c2 with r ≡ 1 mod 4. In this case, β2 = b2 2 and

2 2k−1 2 2k−1 T2 = {1, a2, a2, . . . , a2 , c2, c2a2, c2a2, . . . , c2a2 },

with 2k −2k x, y ∈ F a2 + a2 , satisfying (1 + x2 + y2) = 0.

Proof. Take T = TE/G a right transversal of E in G. The conjugates of by the elements of T are the “diagonal” elements in the matrix algebra FGe ' M[G:E](FE) = M[G:E](FH ∗ E/H) (Theorem 3.6). Hence it it sufficient to compute a complete set of orthogonal primitive idempotents for FE = FH ∗ E/H and then add their T -conjugates in order to obtain the primitive idempotents of FGe. So one may assume that G = E and hence e = and T = {1}. Since G = E ≤ NG(K), we have that G = E = NG(K). f Then the natural isomorphism FGK ' F(G/K) maps to C (H/K, 1). So, from now on, we assume that K = 1 and hence H = hai is a cyclic maximal abelian subgroup of G, which is normal in G and e = = C (H/K, 1). The map a 7→ ξ|H| induces an isomorphism φ : FH ' F(ξ|H|). If G = H, then FGe ' Ý F(ξ|H|), a field. So is the only nonzero idempotent. This is case (1)(i) and β2 = 1 = b20 and T2 = {1} = T20 . So in the remainder of the proof, we assume that G 6= H. Using the description of FGe given in Theorem 3.6, one obtains a description of FGe as a classical crossed product (F(ξ|H|)/F, τ), where F = Fqo/[G:H] is the image under φ of the center of FGe, with o the multiplicative order of q modulo |H|. We also know that F(ξ|H|)/F is a field extension of index [G : H] and that Gal(F(ξ|H|)/F ) ' G/H.

4.3. Finite group rings of nilpotent groups 94 Chapter 4. A complete set of orthogonal primitive idempotents

We first consider the case when G is a p-group. Then G and H = hai, say |H| = pn, satisfy the conditions of Lemma 4.5 and therefore G is isomorphic to one of the three groups of this Lemma. Moreover, H has a complement in G if and only if G ' P1 or G ' P2 and, in these cases, τ is trivial. We claim that in these cases it is possible to give a list of l elements x1, . . . , xpl ofPF(ξ|H|)(p = [G : H]) satisfying the conditions of Lemma 4.17 and 1 −1 −1 that the element [G:H] gH∈G/H gH corresponds to βe and the elements φ (x1), . . . , φ (xpl ) correspond to the elements of Te. To prove this, we will use the following fact: if Fs is a subfield of F(ξ|H|) such that ξp ∈ Fs, i l ±i ξ|H| ∈/ Fs, for i = 1, . . . , p − 1, then trF(ξ|H|)/Fs (ξ|H|) = 0. To see this notice that there ipd i exist a minimal integer d such that 1 6= ξ|H| ∈ Fs, since ξp ∈ Fs. Then ξ|H| is a root of pd ipd X − ξ|H| ∈ Fs[X]. By Lemma 3.1, every Galois extension of finite fields is cyclic. Hence i d F(ξ|H|)/Fs is a cyclic Galois extension of degree p , by the Fundamental Theorem of Galois pd ipd i i Theory. Then X − ξ|H| is the minimal polynomial of ξ|H| over Fs. Hence trF(ξ|H|)/Fs (ξ|H|) = i i −i i [ (ξ|H|): (ξ )]tr (ξ ) = 0. We prove similarly that tr (ξ )/ s (ξ ) = 0. F F |H| F(ξ|H|)/Fs |H| F |H| F |H| Assume first that G ' P1 and vp(r − 1) = n − k ≥ 1 (that is, either p is odd, or p = 2 and r ≡ 1 mod 4). This implies that apk ∈ Z(G). We know that F is a subfield k pk pk of index [G : H] = p in F(ξ|H|). Since a ∈ Z(G), ξ|H| = ξpn−k ∈ F , and therefore, pk i k −1 F = F(ξ|H|) = F(ξpn−k ). Set xi = ξ|H|, for i = 0, 1, . . . , p − 1. If i 6= j, then xixj ∈/ F and −1 hence tr (x x ) = 0. Then the elements x , . . . , x k satisfy the conditions of Lemma F(ξ|H|)/F i j P 0 p −1 e 1 −1 −1 4.17, and hence the conjugates of b = [G:H] gH∈G/H gH by φ (x0), . . . , φ (xpk−1) form −1 a complete set of orthogonal primitive idempotents of FGe. Note that the elements φ (xi) form the elements of T20 if p is odd and the elements of T2 in case (1)(i) otherwise. 2n−2 Assume now that G is still P1, but with p = 2 and r 6≡ 1 mod 4, or equivalently a is not central. In this case ξ4 ∈/ F and F (ξ4) is the unique subextension of F(ξ|H|)/F(ξ4) of k−1 2k−1 i index [G : H]/2 = 2 . That is F (ξ4) = F(ξ|H| ) = F(ξ2n−k+1 ). We take xi = ξ|H| and 2n−2+i i k−1 −1 ±1 x2k−1+i = ξ|H| = ξ4ξ|H|, for i = 0,..., 2 − 1. Hence, if i 6= j, then xixj is either ξ4 ±i ±1 ±i k−1 i ±i or ξ|H| or ξ4 ξ|H|, with i = 1,..., 2 − 1. As ξ|H| ∈/ F (ξ4), we have trF(ξ|H|)/F (ξ4)(ξ|H|) = 0. Since ±1 ±1 trF(ξ|H|)/F (ξ4 ) = trF (ξ4)/F (trF(ξ|H|)/F (ξ4)(ξ4 )) = 0, ±i ±i trF(ξ|H|)/F (ξ|H|) = trF (ξ4)/F (trF(ξ|H|)/F (ξ4)(ξ|H|)) = 0 and ±1 ±i ±1 ±i trF(ξ|H|)/F (ξ4 ξ|H|) = trF (ξ4)/F (trF(ξ|H|)/F (ξ4)(ξ4 ξ|H|)) = 0, −1 we deduce that tr (x x ) = 0 for every i 6= j. Then the elements x , . . . , x k satisfy F(ξ|H|)/F i j P 0 2 −1 e 1 the conditions of Lemma 4.17, and hence the conjugates of b = [G:H] gH∈G/H gH by the −1 elements φ (xi) form a complete set of orthogonal primitive idempotents of FGe. In this e −1 case b20 = 1, T20 = {1}, β2 = b and the elements φ (xi) form the elements of T2 in case (1)(ii). 2k −2k −1 Now assume that G ' P2. Then F = F(ξ|H| +ξ|H| ) = F(ξ2n−k +ξ2n−k ). The same argument k+1 i i as in the previous case shows that the 2 elements of the form xi = ξ|H| and x2k+i = ξ4ξ|H|, k for i = 0,..., 2 − 1, satisfy the conditions of Lemma 4.17. In this case b20 = 1, T20 = {1}, ß −1 β2 = hb, ci and the elements φ (xi) form now the set T2 of case (1)(ii).

4.3. Finite group rings of nilpotent groups 95 Chapter 4. A complete set of orthogonal primitive idempotents

2k Now assume that G ' P3, i.e. H does not have a complement in G. Then F = F(ξ|H| + −2k −1 k ξ|H| ) = F(ξ2n−k + ξ2n−k ). From the first case, one can provide 2 orthogonal idempotents e k e e e e by taking the conjugates of b by 1, a, . . . , a2 −1. Moreover, bFGb = bFha, cib = φ−1(F ) + −1 2n−2 −1 −1 2n−2 φ (F )a + φ (F )c + φ (F )a c ' H(F ). Hence FGe ' M2k (H(F )). Since we know that FGe ' M2k+1 (F ), H(F ) must split, and therefore we have to duplicate the number of 1 2n−2 2n−2 idempotents. From Lemma 4.18, we know that f = 2 (1 + xa + ya c) and 1 − f = 1 2n−2 2n−2 2 (1 − xa − ya c) form a complete set of orthogonal primitive idempotents of H(F ), k k with x, y ∈ F(a2 + a−2 ) such that (x2 + y2 + 1) = 0. Observe that 1 − f = f c. Now define e 2k−1 2k−1 β2 = bf and T2 = {1, a, . . . , a , c, ca, . . . , ca }. It is easy to verify that the conjugates of β2 with elements of T2 are orthogonal, as wanted. This is case (2). Let us now consider the general case, where G is not necessarily a p-group. Then we have to combine the odd and even parts. Since G is finite nilpotent, G = G2 × Gp1 × · · · × Gpr = G2 × G20 , with pi an odd prime for every i = 1, . . . , r. Then (H, 1) is a strong Shoda pair of G if and only if (Hpi , 1) is a strong Shoda pair of Gpi , for every i = 0, . . . , r, with p0 = 2. Take

χ ∈ C ∈ C(H), then one can consider χ as χ = χ0χ1 ··· χr, with χi characters of Hpi . Since

χ is faithful, all χi are faithful and hence we can consider Ci = Cq(χi) ∈ C(Hpi ). Recall that the deﬁnition of Ci (Hpi , 1) is independent of the choice of ψ ∈ Ci. Moreover Y Y X −1 −1 Ci (Hpi , 1) = |Hpi | tr (ξ )/ (χi(hi))hi F |Hpi | F i i h ∈H i p i ! X Y −1 −1 = |H| tr (ξ )/ (χi(hi)) h . F |Hpi | F h=h0h1···hr∈H i hi∈Hpi Since the Galois group Gal( (ξ )/ ) is generated by the Frobenius automorphism x 7→ xq, F |Hpi | F we have Y X X −1 ql0 ql1 qlr −1 Ci (Hpi , 1) = |H| χ0(h0) χ1(h1) ··· χr(hr) h , i h=h0h1···hr∈H 0≤li

where oi is the multiplicative order of q modulo |Hpi |. Since |H2|, |Hp1 |,..., |Hpr | are coprime, o = o0o1 ··· or, and by comparing the number of elements that occur in the inner sum, we have Y X X −1 ql −1 Ci (Hpi , 1) = |H| (χ0(h0)χ1(h1) ··· χr(hr)) h i h=h0h1···hr∈H 0≤l

= C (H, 1). Q Now it follows that FGe = F( i Gpi Ci (Hpi , 1)) ' ⊗iFGpi Ci (Hpi , 1), the tensor product over F of the simple algebras FGpi Ci (Hpi , 1). On the other hand, we have seen that FGpi Ci (Hpi , 1) ' M (F ), for ﬁnite ﬁelds F . Therefore, [Gpi :Hpi ] i i Ge ' ⊗ G (H , 1) ' ⊗ M (F ) ' M (F ), F iF pi Ci pi i [Gpi :Hpi ] i [G:H]

4.3. Finite group rings of nilpotent groups 96 Chapter 4. A complete set of orthogonal primitive idempotents for a finite field F . Hence a complete set of orthogonal primitive idempotents of FGe can be obtained by multiplying the different sets of idempotents obtained for each tensor factor.

0 0 0 Each Gpi , with i ≥ 1, takes the form haii o hbii ' P1 and so G2 = ha2 i o hb2 i, with ki pi 0 0 a2 = a1 ··· ar and b2 = b1 ··· br. Having in mind that ai is central in Gpi , one can easily deduce, with the help of the Chinese Remainder Theorem, that the product of the diﬀerent Ý primitive idempotents of the factors from the odd part are the conjugates of b 0 0 (H 0 , 1) by 2 C2 2 [G20 :H20 ]−1 the elements of T20 = {1, a20 , . . . , a20 } as wanted. The primitive central idempotents of the even part give us β2C2 (H2, 1) and T2, in the diﬀerent cases. Hence, multiplying the Ý primitive idempotents of the odd and even parts will result in conjugating the element b20 β2 by the elements of T20 T2. 2

However one could easily implement this method in GAP in the same way we did for the rational group algebra over a nilpotent group, this is outside the scope of this thesis. One can also easily obtain a complete set of matrix units of FGe. This is stated in the following Corollary.

Corollary 4.20. Let G be a finite group and F a finite field of order q such that (|G|, q) = 1. For every primitive central idempotent e = eC (G, H, K), with (H,K) a strong Shoda pair of G and C ∈ C(H/K), let Te and βe be as in Theorem 4.19 and set E = EG(H/K). For every 0 t, t ∈ Te, let −1 0 Ett0 = t βet . P 0 0 Then {Ett | t, t ∈ Te} gives a complete set of matrix units in FGe, i.e. e = t∈Te Ett and

Et1t2 Et3t4 = δt2t3 Et1t4 , for every t1, t2, t3, t4 ∈ Te. Moreover EttFGEtt ' F , where F is the ﬁxed ﬁeld of FHC (H,K) under the natural action of E/H.

Proof. We know from Theorem 4.19 that the set {Ett | t ∈ Te} is a completeP set of orthogonal primitive idempotents of FGe. Therefore it easily follows that e = t∈Te Ett and

Et1t2 Et3t4 = δt2t3 Et1t4 , for t1, t2, t3, t4 ∈ Te. From Theorem 3.6, one knows that FGe ' M[G:H](F ), with F the ﬁxed subﬁeld of FHC (H,K) under the natural action of E/H. Hence EttFGEtt is isomorphic to F . 2

4.3. Finite group rings of nilpotent groups 97 Chapter 5

Appendix: some GAP functions

In this Chapter, one can ﬁnd all auxiliar GAP functions I wrote and used in some examples. These functions are not optimalized, so it could happen that your GAP system exceeds the permitted memory. #Checks if G is a metabelian group IsMetabelian := function(G) local bool,abnormsubgr,normsubgr,H; bool := false; abnormsubgr:=[]; normsubgr:=NormalSubgroups(G);

for H in normsubgr do; if IsAbelian(H) then Add(abnormsubgr,H); fi; od; for H in abnormsubgr do; if IsAbelian(FactorGroup(G,H)) then return true; fi; od;

return bool; end ;

#Checks if G is an abelianbysupersolvable group IsAbelianBySupersolvable := function(G) local bool,abnormsubgr,normsubgr,H; bool := false; abnormsubgr:=[]; normsubgr:=NormalSubgroups(G);

for H in normsubgr do; if IsAbelian(H) then Add(abnormsubgr,H); fi; od;

for H in abnormsubgr do; if IsSupersolvable(FactorGroup(G,H)) then return true; fi; od;

return bool; end ;

98 Chapter 5. Appendix: some GAP functions

#Returns the smallest non-metabelian group(s) SmallestNonMetabelian := function() local n, groups,list,bool,g; n :=1; list :=[]; bool:=true;

while bool do groups := AllSmallGroups(n); for g in groups do if IsMetabelian(g)=false then StructureDescription(g); Append(list,[g]); fi; od; if list =[] then n := n+1; else bool:=false; fi; od;

return [n,list]; end ;

#Returns the smallest non-abelianbysupersolvable group(s) SmallestNonAbelianBySupersolvable := function() local n, groups,list,bool,g; n :=1; list :=[]; bool:=true;

while bool do groups := AllSmallGroups(n); for g in groups do if IsAbelianBySupersolvable(g)=false then StructureDescription(g); Append(list,[g]); fi; od; if list =[] then n := n+1; else bool:=false; fi; od;

return [n,list]; end ;

#Returns the smallest non-monomial group(s) SmallestNonMonomial := function() local n, groups,list,bool,g; n :=1; list :=[]; bool:=true;

while bool do

99 Chapter 5. Appendix: some GAP functions

groups := AllSmallGroups(n); for g in groups do if IsMonomial(g)=false then StructureDescription(g); Append(list,[g]); fi; od; if list =[] then n := n+1; else bool:=false; fi; od;

return [n,list]; end ;

#Returns a list with data describing each simple algebra #realizable by a strong Shoda pair SimpleFactorsFromSSP := function(G) local QG,SSP,list,HK; QG:=GroupRing(Rationals,G); SSP:=StrongShodaPairs(G); list :=[];

for HK in SSP do Append(list,[SimpleAlgebraByStrongSPInfo(QG,HK[1],HK[2])]); od;

return list; end ;

#Returns a Shoda pair (H,K) such that the idempotent e equals e(G,H,K) IsInE2:=function(QG,e) local grps,l,G,H,K; G:=UnderlyingMagma(QG); grps:=Flat(List(ConjugacyClassesSubgroups(G),AsList));

for H in grps do for K in grps do if IsSubgroup(H,K) and IsNormal(H,K) then if IsShodaPair(G,H,K) then if e=CentralElementBySubgroups(QG,H,K) then return [H,K]; fi; fi; fi; od; od;

return false; end ;

100 Chapter 5. Appendix: some GAP functions

#Returns a list of pairs of subgroups (H,K) such that K is normal in H NormalPairs := function(G) local list,L,L2,H,K,classes; list :=[]; classes:=ConjugacyClassesSubgroups(G);

for L in classes do for H in L do for L2 in classes do for K in L2 do if (H<>Subgroup(G,[]) and IsSubgroup(H,K)=true and IsNormal(H,K)=true) then Append(list,[[H,K]]); fi; od; od; od; od;

return list; end ;

#Returns a pair of subgroups (H,K) with K normal in H and a rational number #alpha such that e equals alpha e(G,H,K) #Returns false if no such pair can be found FindeGHK := function(G,e) local pairs,QG,P,alpha,H,K,e1,bool,i; QG:=GroupRing(Rationals,G); pairs := NormalPairs(G);

for P in pairs do bool:=true; H:=P[1]; K:=P[2]; e1:=CentralElementBySubgroups(QG,H,K); if Size(CoefficientsAndMagmaElements(e1))<>0 then alpha := CoefficientsAndMagmaElements(e)[2] / CoefficientsAndMagmaElements(e1)[2]; if Size(CoefficientsAndMagmaElements(e)) = Size(CoefficientsAndMagmaElements(e1)) then for i in [1..Size(CoefficientsAndMagmaElements(e))/2] do if CoefficientsAndMagmaElements(e)[2*i] <> alpha*CoefficientsAndMagmaElements(e1)[2*i] then bool:=false; fi; od; if bool=true then return [P,alpha]; fi; fi; fi; od;

return false; end ;

101 Chapter 5. Appendix: some GAP functions

#Checks whether N has a complement in G, #if so it returns the complement IsSemiDirectSummand:=function(G,N) local M,grps;

if G=N then return Subgroup(G,[]); fi;

if IsSubgroup(G,N)=false or IsNormal(G,N)=false then return false; fi;

grps:=Flat(List(ConjugacyClassesSubgroups(G),x->AsList(x))); for M in grps do if Size(Intersection(N,M))=1 and ClosureGroup(N,M)=G then return M; fi; od;

return false; end ;

#Returns a complete set of orthogonal primitive idempotents of the #simple factor of QG associated to the strong Shoda pair (H,K) CompleteSetOrthogonalPrimitiveIdempotents:=function(QG,H,K) local eps,G,N,HK,nK,homNK,Heven,Neven,Hodd,Nodd,HKeven,HKodd, NKeven,NKodd,Todd,Teven,TGN,MKeven,aeven,aodd,bodd, beven,ceven,betaeven,g,id,i,j,k,m,n,betae,Te,basis;

G:=UnderlyingMagma(QG);

if IsSemisimpleRationalGroupAlgebra(QG)=false then Print("QG has to be a semisimple rational group algebra: "); return false; fi; if IsNilpotent(G)=false then Print("G has to be a nilpotent group: "); return false; fi; if IsStrongShodaPair(G,H,K)=false then Print("(H,K) has to be a strong Shoda Pair of G: "); return false; fi;

eps:=IdempotentBySubgroups(QG,H,K); N:=Normalizer(G,K); homNK:=NaturalHomomorphismByNormalSubgroup(N,K); nK:=Image(homNK,N); HK:=Image(homNK,H); Heven:=SylowSubgroup(H,2); Hodd:=Complementclasses(H,Heven)[1]; Neven:=SylowSubgroup(N,2);

102 Chapter 5. Appendix: some GAP functions

Nodd:=Complementclasses(N,Neven)[1]; HKeven:=SylowSubgroup(HK,2); HKodd:=Complementclasses(HK,HKeven)[1]; NKeven:=SylowSubgroup(nK,2); NKodd:=Complementclasses(nK,NKeven)[1]; if(Size(HKeven)=1) then aeven:=Identity(H); else aeven:=Representative(Intersection(PreImages (homNK,MinimalGeneratingSet(HKeven)[1]),H)); fi; if(Size(HKodd)=1) then aodd:=Identity(H); else aodd:=Representative(Intersection(PreImages (homNK,MinimalGeneratingSet(HKodd)[1]),H)); fi; if Size(NKodd)=Size(HKodd) then bodd:=Identity(N); else bodd:=Representative(PreImages(homNK,MinimalGeneratingSet (Complementclasses(NKodd,HKodd)[1])[1])); fi; TGN:=AsList(RightTransversal(G,N)); Todd :=[]; for i in [0..Size(Nodd)/Size(Hodd)-1] do Append(Todd,[aodd^i]); od;

Teven :=[]; basis :=[]; MKeven:=IsSemiDirectSummand(NKeven,HKeven); if MKeven<>false then betaeven:=AverageSum(QG,PreImages(homNK,MKeven)); if IsCyclic(MKeven) and (Size(HKeven)<=2 or Centralizer(NKeven,Image(homNK,aeven)^(Size(HKeven)/4))=NKeven) then for i in [0..Size(MKeven)-1] do Append(Teven,[aeven^i]); od; else for i in Flat([[0..Size(MKeven)/2-1], [Size(HKeven)/4..Size(HKeven)/4+Size(MKeven)/2-1]]) do Append(Teven,[aeven^i]); od; fi; else m:=Size(HKodd)/(Size(Nodd)/Size(Hodd)); n:=Log(Size(HKeven),2); k:=Log((Size(NKeven)/Size(HKeven))/2,2); for g in RightTransversal(NKeven,HKeven) do if Image(homNK,aeven)^g = Image(homNK,aeven)^(-1) then ceven:=Representative(PreImages(homNK,g)); fi; od; if Size(NKeven)=2*Size(HKeven) then beven:=Representative(PreImages(homNK,Identity(NKeven))); else beven:=Representative(PreImages(homNK,MinimalGeneratingSet (IsSemiDirectSummand(NKeven,Subgroup

103 Chapter 5. Appendix: some GAP functions

(nK,[Image(homNK,aeven),Image(homNK,ceven)])))[1])); fi; if m=1 or (IsOddInt(OrderMod(2,m)) and n-k<=2) then for i in [0..2^k-1] do Append(Teven,[aeven^i]); od; betaeven:=AverageSum(QG,Subgroup(G,[beven])); else for i in [0..2^k-1] do Append(Teven,[aeven^i]); od; betaeven:=AverageSum(QG,Subgroup(G,[beven])); basis:=[aodd^(Size(Nodd)/Size(Hodd)), aeven^(2^k)]; Print("Warning: This set is not complete! "); Print("One should multiply these idempotents with "); Print("1/2(1+x*a2^power+y*a2^power*c2) and "); Print("1/2(1-x*a2^power-y*a2^power*c2), where x and y "); Print("are solutions of (1+x^2+y^2)epsilon=0 in "); Print("Q(v,w+w^-1)epsilon"); fi; fi;

betae:=AverageSum(QG,Subgroup(G,[bodd]))*betaeven*eps; Te :=[]; for i in Todd do for j in Teven do for k in TGN do Append(Te,[i*j*k]); od; od; od; id :=[]; for i in Te do Append(id,[betae^i]); od; if basis =[] then return rec(Idempotents:=id); else return rec(Idempotents:=id,v:=basis[1], w:=basis[2],a2:=aeven,power:=2^(n-2), c2:=ceven,epsilon:=eps); fi; end ;

#Returns a complete set of orthogonal primitive idempotents of each #simple factor of QG which is associated to a strong Shoda pair (H,K) OrthogonalPrimitiveIdempotents:=function(QG) local G,rc,SSP,P,list;

G:=UnderlyingMagma(QG);

if IsSemisimpleRationalGroupAlgebra(QG)=false then Print("QG has to be a semisimple rational group algebra: "); return false; fi;

104 Chapter 5. Appendix: some GAP functions

if IsNilpotent(G)=false then Print("G has to be a nilpotent group: "); return false; fi;

rc:=StrongShodaPairsAndIdempotents (QG); SSP:=rc.("StrongShodaPairs"); list :=[]; for P in SSP do Append(list,[CompleteSetOrthogonalPrimitiveIdempotents(QG,P[1],P[2])]); od;

return list; end ;

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109 Index

K-character, 20 primitive idempotent, 15 K-representation, 19 2’-part, 79 quaternion algebra, 24 2-part, 79 rational group algebra, 18 abelian-by-supersolvable, 36 reduced, 19 action, 22 reducible, 19 redundant, 42 Brauer, 62 regular character, 21 representation, 19 central simple algebra, 23 character ﬁeld, 20 semisimple, 14 class function, 20 Shoda, 29 classical crossed product, 23 Shoda pair, 29 commutator, 28 simple component, 16 complete, 42 skew group ring, 23 constituent, 21 split, 24 crossed product, 23 strong monomial, 40 cyclotomic class, 65 strong Shoda pair, 33 cyclotomic class modulo n, 66 supersolvable, 36 support, 17 factor system, 22 faithful, 22 twisting, 22 ﬁeld trace, 62 Frobenius automorphism, 66 valuation, 76 group algebra, 17 Wedderburn decomposition, 16 group ring, 17 Wedderburn-Artin, 16 induced class function, 22 irreducible, 19 linear character, 20

M-group, 31 Maschke, 18 matrix K-representation, 19 metabelian, 40 metacyclic, 41 monomial, 31 normalizer, 30

Perlis-Walker, 18 primitive central idempotent, 16

110