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Characteristically Simple Subgroups of Quasiprimitive Permutation Groups

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Characteristically Simple Subgroups of Quasiprimitive Permutation Groups UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIENCIAS^ EXATAS CHARACTERISTICALLY SIMPLE SUBGROUPS OF QUASIPRIMITIVE PERMUTATION GROUPS Pedro Henrique Pereira Daldegan Belo Horizonte - MG 2018 Pedro Henrique Pereira Daldegan Characteristically simple subgroups of quasiprimitive permutation groups A thesis submitted for the degree of Doctor of Philosophy of the Universidade Federal de Minas Gerais. Supervisor: Csaba Schneider Universidade Federal de Minas Gerais March 2018 Agradecimentos O correr da vida embrulha tudo, a vida ´eassim: esquenta e esfria, aperta e da´ıafrouxa, sossega e depois desinquieta. O que ela quer da gente ´ecoragem. O que Deus quer ´ever a gente aprendendo a ser capaz de ficar alegre a mais, no meio da alegria, e inda mais alegre ainda no meio da tristeza! S´oassim de repente, na horinha em que se quer, de prop´osito { por coragem. Ser´a?Era o que eu `as vezes achava. Ao clarear do dia. Jo~aoGuimar~aesRosa Quando eu terminei o mestrado, o doutorado n~ao foi um passo imediato. Ansiava por come¸carlogo a colocar em pr´atica as coisas que aprendi. Contudo, ao perceber que poderia simultaneamente fazer pesquisa e lecionar no departa- mento, encontrei o meu lugar. O doutorado se mostrou um desafio muito mais dif´ıcil quando comparado ao mestrado, pois me foi (e ainda ´e)exigido a todo instante desenvolver qualidades de pesquisador, o que torna o trabalho por vezes muito solit´ario e estressante. Gostaria de expressar ent~ao a minha gratid~aoa todos que contribu´ıram para a elabora¸c~aodessa tese, seja com conversas ´uteis ou in´uteis, ou idas ao cinema, ou como companhias para o v^olei ou para dan¸car. Afinal de contas, nas entrelinhas, escoando das minhas m~aos para o teclado, houve muito ch´a,cinema e samba. E sempre haver´a, am´em. Bem, nada seria poss´ıvel sem o meu orientador, Prof. Csaba Schneider, que me introduziu no estudo dos grupos de permuta¸c~oes, e que sempre me apoiou, pressionou, e compreendeu nos momentos dif´ıceis. Obrigado por sua paci^encia e aten¸c~ao, por sempre me manter em movimento, e por sua dedica¸c~aoao nosso projeto. Agrade¸co aos professores e funcion´arios da UFMG, em especial `aProfa. Ana Cristina Vieira, que mescla carinho e firmeza, e que sempre acreditou em mim, `asmeninas Eliane Kelli e Eliane Andr´ea, que me aguentam na secretaria e fora dela, aos Profs. Rafael Bezerra, John MacQuarrie, R´emy Sanchis e Michel Spira, obrigado pelas conversas e pela amizade. Agrade¸co aos membros da banca, Profa. Ana Cristina Vieira (UFMG), Profa. Carmela Sica (UFBA), Profa. Cristina Acciarri (UnB), Prof. John MacQuarrie 2 3 (UFMG) e Prof. Martino Garonzi (UnB). Agrade¸co `aCAPES e ao CNPq, o apoio financeiro. Agrade¸co carinhosamente aos meus amigos, `aLilian Batista, Lorena Oliveira, R^oney Rachide, Daniel Ungaretti, Marina Muniz e Jeanne Amaral, que bem sabem o que ´eser matem´atico, ao Pedro Oliveira, Renata Soares e Thalis Diniz, que desconhecem a medida do tempo, `aK^enia Ribeiro, Frederico Santiago e Luana Ribeiro, que est~aosempre de prontid~ao, e a todos os outros que ir~ao brigar, com raz~ao, por n~ao terem seus nomes aqui explicitados. Voc^esfazem da minha vida uma jornada muito alegre e mais leve. Guardo voc^escom muito carinho. Ao Esdras Mellemberg, que chegou tarde e sem avisar, bagun¸coutudo, e ilumina os meus segundos. A voc^ededico o meu amor. Finalmente, agrade¸coaos meus pais, meus irm~aos e meus familiares, que tanto me incentivaram (\Pra que foi ca¸car fazer matem´atica. V^ese pode, minino!") e torceram, mesmo sem entender. A voc^esagrade¸coos cafun´es, as alegrias e preocupa¸c~oes de todos os dias, que fazem eu me sentir mais vivo e comprometido. Muito, muito obrigado... Abstract We say that G ≤ Sym(Ω) is transitive if G has just one orbit on Ω, namely fΩg. If G is transitive on Ω and the only partitions of Ω preserved by G are fΩg and ffαg: α 2 Ωg, then we say that G is primitive. The O'Nan-Scott Theorem [25] classifies the finite primitive permutation groups by dividing them into classes, according to the structure of their minimal normal subgroups. An important result in this classification is that every permu- tation group admits at most two distinct transitive minimal normal subgroups [8, Lemma 5.1]. A permutation group is quasiprimitive if all its nontrivial normal subgroups are transitive. For example, all primitive permutation groups are quasiprimitive. Finite quasiprimitive groups were characterized by Cheryl Praeger [28], who showed that they can be classified similarly to the O'Nan-Scott classification of finite primitive permutation groups. The inclusion problem for a permutation group H asks to determine the possible (primitive or quasiprimitive) subgroups of the symmetric group that contain H. In other words, given a permutation group H < Sym(Ω), we are asking about its overgroups. For instance, it is a common situation in algebraic combinatorics that we know a part of the group of automorphisms of a com- binatorial structure (for example, a Cayley graph) and we wish to determine a larger automorphism group which may be primitive or quasiprimitive. In this work we describe all inclusions H ≤ G such that H is a transitive non- abelian characteristically simple group and G is a finite primitive or quasiprim- itive permutation group with nonabelian socle. The study of these inclusions is possible since we have detailed information concerning factorizations of finite nonabelian simple groups. For this reason, many of the results presented here rely on the classification of finite simple groups, specially chapters 4 and 7. 4 Resumo Seja Ω um conjunto finito n~ao vazio e considere G um grupo de permuta¸c~oes de Ω, isto ´e, G ≤ Sym(Ω). Dizemos que G ´e transitivo em Ω se dado qualquer par de pontos em Ω, existir uma permuta¸c~ao em G que associa estes pontos. Por outro lado, dizemos que G ´e primitivo em Ω se G for transitivo e n~ao preservar uma parti¸c~ao n~aotrivial de Ω { por parti¸c~ao trivial queremos dizer a parti¸c~ao formada por subconjuntos de um ´unico ponto e a parti¸c~ao composta somente por Ω. Por exemplo, o grupo sim´etrico e o grupo alternado de Ω em suas a¸c~oes naturais s~aoambos primitivos. O Teorema de O'Nan-Scott classifica os grupos de permuta¸c~oes primitivos finitos dividindo-os em classes, de acordo com a estrutura de seus subgrupos normais minimais [25]. Um importante resultado nesta classifica¸c~ao´eque todo grupo de permuta¸c~oes admite no m´aximo dois subgrupos normais minimais tran- sitivos distintos [8, Lemma 5.1]. Um grupo de permuta¸c~oes ´edito quase-primitivo se todos os seus subgru- pos normais n~ao triviais s~ao transitivos. Por exemplo, todo grupo de permu- ta¸c~oes primitivo ´equase-primitivo, mas visto que um grupo simples transitivo ´e sempre quase-primitivo, mas nem sempre primitivo, a classe dos grupos quase- primitivos ´eestritamente maior que a classe dos grupos primitivos. Grupos de permuta¸c~oes quase-primitivos finitos foram caracterizados por Praeger em [28]. Neste artigo Praeger mostrou que os grupos quase-primitivos podem ser classificados similarmente `aclassifica¸c~ao dos grupos primitivos. O problema de inclus~ao para um grupo de permuta¸c~oes H almeja determinar os poss´ıveis subgrupos (quase-primitivos ou primitivos) do grupo sim´etrico que contenham H. Tal problema possui um n´umero importante de aplica¸c~oes em teoria de grupos, combinat´oria alg´ebrica e teoria alg´ebrica de grafos. Por exem- plo, ´euma situa¸c~ao comum em combinat´oria alg´ebrica sabermos uma parte do grupo de automorfismos de uma estrutura combinat´oria, por exemplo um grafo de Cayley, e queremos determinar um grupo de automorfismos maior que pode 5 6 ser quase-primitivo ou primitivo. O estudo do problema de inclus~ao ´eposs´ıvel visto que h´ainforma¸c~oes de- talhadas sobre as fatora¸c~oesde grupos simples n~ao abelianos finitos. Alguns resultados gerais sobre tais fatora¸c~oes podem ser encontrados em [1, 26]. Neste trabalho descrevemos todas as inclus~oes H ≤ G tais que H ´eum grupo caracteristicamente simples, n~aoabeliano e transitivo, e G ´eum grupo de permuta¸c~oesfinito quase-primitivo com socle n~ao abeliano. A este tipo de inclus~ao damos o nome de inclus~ao CharS-QP. Tais inclus~oes ocorrem naturalmente, por exemplo quando tomamos um grupo quase-primitivo finito G que possui um subgrupo normal minimal n~ao abeliano S. Se denotarmos por soc(G) o socle de G, tanto S ≤ G quanto soc(G) ≤ G s~aoinclus~oes CharS-QP. Ao tratar esse problema, nosso primeiro passo foi imergir explicitamente, sob algumas hip´oteses, um grupo de permuta¸c~oes quase-primitivo em um produto entrela¸cado com a a¸c~ao produto, de forma que tal imers~ao fosse permutacional. Teorema 1. (Teorema de Imers~ao)Sejam G um grupo de permuta¸c~oesquase- primitivo em Ω e α 2 Ω. Assumamos as seguintes condi¸c~oes: 1. S = Q1 × · · · × Qr ´eum subgrupo normal minimal de G, em que cada Qi ´ecaracteristicamente simples e n~aoabeliano, e r ≥ 2. 2. G age transitivamente em fQ1;:::;Qrg por conjuga¸c~ao. 3. Consideremos as proje¸c~oes πi : S ! Qi, e assumamos que Sα = (Sαπ1) × · · · × (Sαπr): Se considerarmos Γ := [Q1 :(Q1)α], ent~aoexiste uma imers~aopermutacional : G,! Sym(Γ) wr Sr, em que consideramos o produto entrela¸cado como um grupo de permuta¸c~oesagindo com a a¸c~aoproduto em Γr.
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