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Gs(k). order of be harder. be and blocks the 4513-4547 the OF about We (Brauer) Hall Florida a of 2-block cyclic USA 5. Characteristics way the algebraic 604800 Loewy THE the of shall In of the group the ([16], 2 Hall-Janko that particular, defect will have G hc was which characters A (1996) determine In has HALL-JANKO Hall-Janko projective modules. projective structure = of of Gz(2)' most our [12], eventually group not the been 27.33.52.7 maximal are group, approach, Hall-Janko we [13]), of yet 2 3 described in described G2(k), treated of group and studied the the PSU(3,3), determine of been simple are as the furnish defect, irreducible extensions 5 and also and have are where studied we GROUP in projec- in group of mod- rela- [15]. The rely one [17] the the its [8] In k Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 interchanged module THEOREM. highest P module ules from (see For twisting investigations between this may modules to acting character-theoretic. Unfortunately, our Unfortunately, character-theoretic. Z(G) 4514 G There are Let is of an p 52 direct with be also their the group the = algorithm. L L on below) quite weights. and extensions 2, [E, by be by two subscript the suitable projective the approach, it approach, twists the GI. indirect two conjugacy In a simple a the by challenging. projective simple modules simple of so k6-projective the outer automorphism outer column labelled column extension the other The that Nevertheless, by TABLE are 1 tables methods, Frobenius the are kG-module nodules (for groups. the same; this is cover the Frobenius necessary first necessary In a classes below 1 1. fixed modules restrictions this paper, -+ cover for of The is with map we z(@ by L h entry the which and embedding the of hope P principal ,. when a. This map. embeddings the (see subscripts of of and dimension P follows + the G. second. collection L we that to its of [18]). both G this is find simple initial calculate of projective 2-block -+ different of some from first the We 2 are G turns matrices G of of are then data in G-modules of o labelled row G -+ have chosen have of step regarded 5-term h space the of Gz(k)) the tricks into them from out 1, cover. needed G. is extensions using of GZ(4) not to the sequences P does not does will group generators group the as be but the but with automatic If restrictions are of the kG-modules. our notation our be and simple the p by ,-. extensions essentially # useful for useful restricted fact of the they same as same amount 2, arising simple simple mod- then first that and SIN are to Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 HALL-JANKO p block regard picks are complex 361 = to o self-dual, not 3. out of be to TABLE Our notation Our maximal conjugation the the one one pair GROUP of 3. the for of eet Since defect. The TABLE we dual which two 36-dimensional two fixes follows can non-principal simple modules all E~t:~(61, choose 2. that other The these of simple Brauer principal either [9]. 361) 3-block two There modules.) of are to # dimension 0. 3-block be of the is (See 6 the one characters. only simple only of below of one maximal choice 36 G. in labelled that the to We modules defect. be this non-principal have chosen have made with made 361, condition which since 4515 Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 p=5. modules in these modules in Morita a complex which a which Rickard the derived equivalent REMARK. Again same gives TABLE qiaet since equivalent, has our notation matches our notation integral Notice a provided change 5. ought two quadratic The in that TABLE a some lcs which blocks of particularly non-principal to there basis they be 4. good evidence good form. a The exists have to tilting [9]. show Also principal simple preserves different a 5-block complex ieto between bijection each that way. to 5-block block the of suggest extensions. Cartan for In two particular he has the of of maximal that the that Cartan matrices. above 14 G. complex The the equivalence matrices two blocks sets defect. has However, characters blocks of described are simple define and not SIN are J. Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 qiaet over equivalent of all HALL-JANKO effective etd for pected These p tic treat low provide published zero, p oue in modules cipal block cipal there factor containing 3 same H simple denoted modules, superscript 3' the REMARK. = = Here H affords H G @ splits of 7 appear 2. 3. scalar S2(3) which case is covered is case has the ere o divisible not degrees F containing table is results of we a little doubt little U3(3) additional into index other 3 convenient reference. convenient the or give The this matrices, -t @I the except of as are the "*" TABLE not. two GROUP 3*, are dimensions 3. isolated calculations, isolated natural has tables 2 isomorphic 27-dimesional a suitable a two extensions smaller denotes duality. denotes principal the in that obtained nonisomorphic, 32-dimensional modules. simple projective The that technique. w blocks, two primes. G2(2). by that Al. 64-dimensional and of is ersnain and representation main the the group, extensions the by The ot r all or most block 1, 5-adic ring. the by for by to entries The theory 6 their As reason for including for reason quotient 5. principal H and similar Steinberg The the oue 15 module the the far of latter So The form dimensions principal on 14. of H. for group order as consideration it scattered throughout scattered principal Steinberg module, Steinberg blocks module of the methods, of is The Their restrictions Their a has I the 2-block am unique class unique the possible module. the leading diagonal leading is group three other 6 aware, block of others are others space the in is 7 block defect of is is except the its simple H that block kernel 25.33.7, this the The of H of of several table ymti square. symmetric = n a and E induced of G2(2) one nontrivial matrices the U3(3). appendix that, principal (G2(2))' also known, also of modules in modules maximal whose of to below. the literature, and so (see blocks Gz(2) are of block H the are as the The the modules are 1 [lo]). restriction of ih be might aua map natural ie by given instead composition is characteris- The block are is subgroups. of results be- results subgroups the trace simply the of whether defect We derived module simple defect has is prin- 4517 0 of The will ex- the an by to to 0. 8 0 Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 complete of Preliminary REMARK. able modules. of LAS, weights simple indicate as find Hornkc(-, X ules bol Chevalley 2.1. In The In In calculations characters which the [X the considered what $2 and the [18] following calculation modules, or : in we correct correspondences correct S] what Table are it follows, In group [5] -) will assume is this isomorphic etc. tables in Remarks. and information which shown A2 yields A2 denote will notational a G2(4) case, TABLE we of in few Extic(-, in projective modules, projective in k of be multiplicities will [9], follow to that a places Brauer characters, Brauer the of vector be to little the 1161, often make often A2. rational points rational is number an conventions the G directly -) used Loewy for extra is The lerial closed algebraically 1121 spaces simple module to among isomorphic p in Hornc(-, of principal and of = work structure rmti numerical this from use each 2. simple composition factors over will [13]. the To of over beyond will decomposition instance. Some care Some instance. a information be used be notations avoid 3-block to -) field modules We of be Fs S. a and the omitted, the will il of field inside maximal subgroup maximal k undue in projective and of Ext&(-, calculations needed also of in tensor in this paper. this H. drawn the these matrices, products we length, characteristic information, of but we use algebraic the will -). from sources. the indecompos- is products kG-module the abbreviate will needed The All tables the details of try to try group mod- syrn- such AT- 2. SIN the to of of to Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 by HALL-JANKO Gz(k). (see Steinberg module Steinberg G; denote subscript stabilizer G2(4) for to fours will must all 141, the cations. module, has form checked are 2.2. sequence This and LEMMA ~~1 PROOF: (ii)), the REMARK. Then The In spite 62 (ii) 61, indeed, simple modules in modules simple G2(k). (i) equation self-dual. dimension ae use make G [Ill) 142, n result @ This group, split, H1(G, and the ~~2 do = H1 structure the 141. The from tables from with hypothesis 1. 36 We in (L1, splits upon are (L,, "2". 5 of 14-dimensional Under G subsection contains a contains subsection There all algebraic conjugates under conjugates algebraic is is Gz(4) := which Let dl V) (Remark: the containing will is simple GROUP vG not the Lz). V) a variation a but The of + 61 d; maximal = section F of d2; show is = the + = 8 it trivial module, a module, trivial zero 641. 0. 641 of Set is just simple modules simple 0, be the 62 1 0. (i) later. of G2(k)-modules a given restriction qiaet to equivalent for all for the for remain irreducible remain and follows certain hermitian Let that These is d, a Since Brauer non-principal 641, one other heading, in equivalent field, of = i adjoint principal Lo, 84 The O-+V-+E+F+O conditions = 14i G2(4),the every dimF the 642 i. the modules 1,2. := from L1 characters. @ Since a G facts main result main to and two the 6, general 142 and short vL' module block, L1 in @ 6-dimensional to block (i) EG group and group whose field a simple module simple a ELO =" tensor La . ocrig ipe modules simple concerning last block on the 142. the do remain irreducible upon restriction irreducible upon remain by and Gal(F4/F2) Suppose exact 36 form with # besubgroupsofG lemma and lemma = in for _> are counting 141 sentence principal @ Note also Note The Lz, dl 0, ihs weights highest has the products in EL' subgroups L, 160. the sequence defect restrictions on +d2, and since [I]. module hence also hence following V been 6i + subgroup module, the finite-dimensional a or implies 8 EL2, dimensions. of block group that VG some 142.) studied are to 160, these 64-dimensional of 84 M~O the FG-modules lemma is = not all simple modules simple all of are is (excluding to isomorphic that with which which immediate appli- immediate We Gz(2)' 0. modules simple also are fact Lo. isomorphic for isomorphic then in = see Lo G ML1 By [15] 2-restricted isomorphic that appears Thus, we can of is then arbitrary. k, 5 modules (ii), the (ii), the +ML2. and G. denote 61, ih a with LlnLz (i) to the (first) all EG of 4519 that FG- E~' full and We 62, the (I) we be to in = Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 .. In 2.2. 4520 classes. actions to itormalizer $2. replace follows 3.PGL(2,9). element but eeain can generation subgroups will routine sending from in will G point composition we As, is whose Let 2-dimensional Dlo were PROOF: 2-dimensional. LEMMA 14y(r) We Let From easy The and As the (b) (a) fix Let w etc. need elementary mainly = the not which will now will x denote z Vl examples restrictions nontrivial simple nontrivial have 61/~(r) uniserial with uniserial 14q~(.) = to (x).2, order on C(z) x then to The Zs, x of h caatr table, character the be These 2. This characters to to be uniserial, 147. check to these modules. these we check C(x) N(x) As be an make be the intersection requires an notation factors. from by that both to do we will mostly modules But concerned the perform will element modules Let element 2 of second that arguments its at in (vl E that denote use root. (X choose orders to course one on a the then A5 not class derived so, it S that further @' primitive dimk(l4f) factors k, factors (z) follows Since of €3 we would = L1 x k) end we factor being generated being factor be Vl some of As of k) certain for the k(SL(2,4))-modules S by (x, Dlo 141 5A are are and @ for with that which be We will follow will We difficult class must = C(x) $ and modules of suffice x') subgroup C(x) Dl (V2 141 same IN(.) discussion is but have w from about (k L, (x, we of calculations with small modules small calculations with read L2 [2]. 5th hypotheses (i) hypotheses C(x) + be Vl V2 €3 and about representations facts about 5A. = €3 is (x). in applying in x') pick an of outside see Z w4 has generate , 02); would Dl) 2, to root self-dual, (Brauer) character for as and facts and off k. order element SL(2,4) to Z D2 for Then and so that and be suitable subgroups in a 3.As. our Z5 from introduce @ of hi tensor their direct if the the the hold semisimple. (Vlz the the structure (x), wZ X C(z) x 3 unity (xz) its needs.) G Z5. subgroup of (in irreducible Likewise, discussion before discussion Lemma 1. In 1. Lemma were x the + so they ATLAS are centralizer and with then 8 summand and Dlo, class by w3 is generate must ls 3A). class in Dz), (The list will two semisimple, x. This x. can respectively. its that k (ii). x' C(x) These we 5C product and of algebraically conjugate in X notation own be where Then statements be calculations. C(x)' if of have: Lo has In 2-dimesional aia subgroups, maximal place X x' e Therefore, be. fixed C(x), and the found, G, L, X has verifying all are centralizer We 5 is trace of in 14r(x) X 2 and in k character V12 Ll C(xl) cases taken of we As index for the typical using for (b). with SL(2,4) is have fact for x since x n w := study would isomorphic concerning = VZ the conjugacy lemma. Lz, + It would it The instance, these to be the generate SL(2,4), Vl we 2 C(z) Lemma @ w4. module is will rest and of in so of VZ, @ their of right clear have may they only if SIN the the For (x) V2. we we an be G, be of X it It is Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 HALL-JANKO are and with belong fixed yields we LEMMA by since notation to p. LEMMA By by PROOF: 36 that H1(L1, 61 hypothesis notation 6(") and SL(2,4) claimed say. also Lemma PROOF: Since Lemma (b) (b) We (a) Pursuing this method this Pursuing (c) (b) @ (a) L1. g 1831) Vl. have some conjugate Lemma We obtain N(xl) 62. = To V2 141 H1(G,6i) 61 x points next apply these facts in a simple a in facts these apply next dl the H1(G, ~xt&(l41,142) H1(G, H1(G, H1(G, Therefore to Now 36) 0, 862, may and @ 2(a), 1 3. prove nonsplit 4. Since 36(~) and x of of = Equation in (a) 6,. the which implies which V2 result. holds (x), generate % H1 the Lemma 0 to the 2(b), x' it we consider holds. It it 36) is 61') 6i 6i) 14i) GROUP the H1(Ll principal A in E (Lz, is Hom~(k, be the are does not does preceding discussion preceding Lz follows an # find principal @ Brauer because G, a fact E Vl = the extension k(Ll/(x))-module we checked = 6;) 0 = M) conjugate in G. in conjugate elementary calculation in calculation elementary lemma 2.1(1) k. 0. leading 1) @ /(x), using Lemma using Vl 0. It G. by C(z)' hypothesis see just = V2 = that , = about that block. suffices a simple 2-dimensional module 0. showing 61 character matter 0. This further, that 36(")) we 0 E that block shows it 141. is @ from dZ and olw. Let follows. to the VI2 H1(Ll/(a), see will that 141) SL(2,4) proves = Let H1(L2,61) dl to We has = whether Lo the do from (ii) Since that suffice that we = 2(a) Z L1 0 check contradiction H1(LI, calculation take = We = composition long and dz and of HomG(G1, M(') have that = (st). 1 that H1(Ll that chracters (b) there that as Vlz lemma = have, using Lemma using have, and M(")) C(x), to M L1 of hypothesis similarly case. we exact = M) the 1 follows X must Lemma 1, Lemma is show be = also From and consider 0, is SL(2,4). an is uniserial. is following. modules for C(x), L2 = = shows 1 such sequence. from 141) nonsplit a that that that be is factors do injective 0. H1(Ll/(x), that /(x), = from characters for verified. a nonsplit extension nonsplit a = C(xl) an = L2 (ii) which 61 dl we 36' ht the that 14:: L2, H1 2, 0, Vl) for extension, = (a), or = 61 of choose the (G, so The which module, = # and C(xt) SL(2,4) Ll 9 0, extension 62. The Lemma (twice), it 2(b), Hypothesis the 0, as 160 does 160 as 0. M(")) M) k existence we as / only follows (x) Lo We because component (see [2] (see MG L1 So Lemma So isomorphism condition a see can and is = = of 2 first we remaining conjugate = 0. = k 1 S. SL(2,4). = k Lo of Ll C(x) for (twice) (in obtain or 0. In by 0. of prove Since N(x) (i) / k = 452 that of (x) not [14, the the the We L1, 61, By by on S. of of g k I 1 , Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 the natural H1(SL(2, The (c) will follow will we 0. tain second he situation this showing 4522 into G group) socle completes knowledge given LEMMA 2.3. sufficient in PROOF: top 141). does (a) nondegenerate see G2(2) The Let and bottom H1(G, SO(6, last To make To layer. in Inside (for to sequence E composition not the H1(G,E) 1. 4)) (a) group corresponding to SL(4, be example the from A2(61)) k), of exact preserve any preserve k) We show proof G is a is G A is equivalent is certain the proof. which is impossible is which = further partial (a) have k) and s easily is trivial of 0. sequences that preimage in preimage of since = = by subgroup if Therfore Lemma Therfore (1). factors 2.2, modules. 0. we the 0. self-duality) description the use G composition 61 It to Lemma nonzero Since can Now seen following has no subquotient will is to of long of simple, (s'(61))~ show L 2.2, 61 therefore S2(61) E to S2(61)/~ 861 roots, 4(c). faithful 4-dimensional quadratic is of SL(3,2) be as Lemma that that module factors the structure an so the latter are 1 zero of To complete M so applies A' we extension = be the k k of 1 (61) L 0. (twice), 1 (generated would using enough 5 is form extension does not does extension structures 61 lies we 141, This uniserial ~'(6~) 1 861 and is need inside k@ isomorphic H1(SL(2,4), obtain on we of means of 141 the to obtained (b) 141. 61 61. k are under to show by (twice) proof @61 (with is proved. is where a by representation. have an Such By we the maximal subgroup maximal reduced 62, H1(G,S2(61)) has embedding the must (b), of by (as root factors Vl) more each and a so (a), already split. In split. omitting form map we an the show subgroups by r 62. it row thus abstract detailed 141, k will in (b) would result From of This been that and is SIN our ob- the the 62, be to = G a Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 HALL-JANKO where of where 3 would it with proved. will and summand mediate be with contradicting quotient have (b) so a simple a PROOF: a REMARK. LEMMA x follows 2-dimensional We G the (h) (b) (d) (a) (g) the (e) (c) (f) 3 (i) The it reappear isomorphic factors matrices to composition can is easily seen have socle V W only simple only ~xt&(61,84) ~xt&(84,84) ~xtL(141, Ext&(36,36) Let Ext&(141,36) complementary exists Ext&(6,, H1 H1 by be semisimple be composition that the 2. is is socle, (a) is that (G, (G, since it belongs it since now 2.2, From M the also the V*, From no GROUP 61 U) often H1(G, of be if natural Lemma establish both simple 8-dimensional U) 8 isomorphic length uniserial to the = trace subspace the V, V*. V, module 141) a factors 141) submodule 0. that nonsplit = we Gz(2)'). characters, = = = 61) factors are natural isomorphic module M module = 0. = 0. zero. 0. and 0. reserve summand. module = 2. 4(a)). 0. r the 0. the Thus, U k 0. kG-module However, k to of then (twice) in direct to Thus, both xeso of extension (2.2, Lemma (2.2, triviality A split surjection split a the of (b) the 61. Then for different iet calculation direct we easily checked. easily MIL U were to U 8-dimensional is summand We notation would have would L. It and must see 61. we isomorphic does Ext&(61, with follows kL-module o uniserial, not Since of have have that Then 62. 61 block I-cohomology k be 4(a)). not factors and by If U (61@16~),~ ~~(61) 6]. V@V 61 u(") that U' U' have 61 for from M) k Also, 141 8 to By @ space were := (unique isomorphic shows it. ~'(61) 61, k = ~~(61). U of = E are socle self-duality, every self-duality, the then in rad 64] has a simple head simple a has this u(~") k 0. k, S not uniserial not its semisimple for in u('). $141, other U/ that 61. splits (V@V*)@(V$V*), length 2 length -+ In its module up many second socle second Since soc and A3(61), particular restriction to to 61~ factors. We we off U isomorphism the cases. this that Z is self-dual is is have as will and uniserial V space it it a direct a module simple L this $ Let would is im- is layer, apply (a) there 4523 to and V*, and of U L is is Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 prove is Now hold block. ML1 u(') 4524 Lz would follows (c) hypothesis Hence holds. now extension central 2.2, O+k-+M*-+61 (b) L1 where an H1(L1, and dimensional again This 2(a)1 known (dl zero. / We = ascending filtration ascending (2) This First 2.2, Lemma follows with also reduces Vl C(xl), the = one d2 Y imply shall By 61 structure % from A @ 0. follows Lemma is a is stronger yields = we SL(2,4), @ Lz straightforward conjugacy X calculates (i) 3 apply M) If using modules Lo 1 show 2.2, us E MG nonsplit in as of not, with Vl H1(L2, = vanishes. to place well. from 2.2, of 3(a), of Lemma 2.2, statement 2.2, f S. @ facts so ~xtL(61, +O. we its Ag L1 of 0, that Lemma V2 over First, all Character computations of (a) Lemma extension Lemma projective modules (see projective modules we U) ol have would L1 by with contrary @ about = L1 summands and dl By 4(a), k, computation are V2 = and (z), C(x) it is easy is it as they as U) that C(Z)' = 0. factors 1 % the Lemma 1 left 2(a) k(SL(2,4)), is L2 and dimk(61 using to It = V12 of to E satisfied. H1 above description it remains the to 0. must act must k Lo that IdL' @ (L1, the SL(2,4) are are suffices M, by show to From the module Vz l(b), = using conjugate @ injective definition 61 check Vl. 62, (xz). and = long M) 8141) which to The that trivially since Lemma to MS, Using M*, x L1 2.2, show 61@M. VIP show that check Since (x), exact sequence associated sequence exact [2] triviality ~xtk(61, = and the are 641 with but Lemma is or and of this dimk H1(Ll, of MS H1(L1 L2 do l(b) an on As (ii) easily [14, M. is We MiL, then both = M and = injective u('), not has HornL, for of p. 6, so we M*) result choose From /(I), 2 C(Z)', k. as U) read the the and 2.2 shows L1, contain 1831). in no see so in (c). in We = hypothesis = the (61 Lz-action nontrivial (61, Lemma follows. namely same do the off 0. 2.2, L1 that module 0, which claim @14~)(~)) that principal = We from M) but = claim S. Lemma d2 By would U C(x), shall This that = that 2(a) is and this has SIN the for (c) on (i) to 2- 3. it a Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 HALL-JANKO so ~xt&(6~, is this Since there and using Lemma using where (e) see first (f) module for module (g) zero using zero use rm Lemma from (h) (i) The 141 To uniserial we Dualizing This We We that This of $ would only summand structure is shall show HomG(62, k the U1 have have a is (proof the follows U') one nonzero = immediate self-duality violate the aim with EX~&(~~,U) G2(4), last by the = U GROUP k, so k, l(a). l(b) of long to is zero is of 0. Lemma and U') from group factors k, factors all exact show the the 2.2 kG we for Suppose is exact We = the U2 injective. from last Lemma (g) map are of 0, since the sequences is modules occurring these modules in claim l(b), is = the some modules, isomorphic and sequence statement done third its latter. (e) A 62, 0, we 641 sequence -+ 4(c)) that Galois since we k. have by 2.2, by and the and @ isomorphism and k. in 642, First filter By in conjunction in Extk(14~, the The and E~t&(6~, of an to conjugate. splits ., Lemma 2.2, being Lemma (c) proof we Extk(641, U extension second we ~xt~(14~,61 kernel (with see by have if the of 62 S2(61)*) factors and that has 3(a). summand 2.2, Lemma 2.2, @I 1 restriction By (d) 141) 642) with replaced by replaced 0 only sequences A 2 3(a), -+ composition osdrn blocks, considering 8 for cannot 61, Z $ (a) U' if 61) ~xt&(14~, the Ext&(ul, it k. Hom~(62, U', can -+ and 2 suffices 4(c) of be Since vanishing. can A Ext&(61,61 61, 2). e seen be the (d). uniserial, and -+ be factors Therefore where Steinberg U2). ~'(61) 61 62 to A) read making @ to -+ show 4525 # The 62). U' we 61 off be as 0, 0. @ Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 4526 from quence for quence ExtG(l41, LEMMA of Using of the ht the that 141) involving Extk(141, ~xtL(36,84). section reverse only reverse (i) 2.4. These so there We exact sequence exact have from We PROOF: to the d 5 From We any (ii) fxt&(14~, quotient have now dimk = By top dimk the is the need will by nonzero direct and leave 0, a 2.1 2. apply making last The HomG(61, to tensor structure ~xt&(61, structure kG-module S2(6,)* -) ~761)) by Hom~(61, ExtG(l41, turn (iii) to we We bottom of after only sum 62) to first map (d) and know Ii Hom~(61, see kG-map have out have products, it the use E the by we inequality of 1 established 36) in of 0 E of H1 in to second sequence second some more ExtL(141, d of ~'(61) + (141 following the have -) the E 86) 61 (2) 6i (G, each copies = Lemma be equal; be group automorphisms group with @6i 36 61 applied e. following @ same -) 8 is 84), 1 found @ -+ 61 summand is not is 61)d) follows o zero, not again, HomG(G1 and socle 61 of E by a (2.3, extensions (ii) k l(b), dimension about -+ @ 141. nonsplit to the $141) we e. consider the difficult = 61 isomorphic ExtL(61 module . One k. we from Lemma prove d. in our Tensoring t claim are the see structure @61, hence first Then the Then (I), = (141 2.3, Lemma extension claim of that to 0. the e. the also , structures can E), resulting simple modules 366, l(a)). sequence in check @ injective Let radical to resulting and easier the 611d with sees follows which be of 62 (iii) long exact long d unique that various some modules. small of realized By and be -+ from inequality 61, l(a). fxtL(141, in layers. 141 where can by 0. 2.3, the the from long El we (1). the structure the nonsplit by the be By natural results dimension soc Lemma as obtain to exact 62. sequence the long exact long o we Now seen definition claim. be the E now 84) This rows isomorphic computed: of extension to identities sequence. kernel the and l(b), and the and the be shows Thus, apply of going yields short that of zero (iv) last SIN se- (i). we Ii dl Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 62 subquotient unique and 2.3, Lemma 2.3, submodule. factors, and PROOF: statements (62 (hence (b) The 1, HALL-JANKO we (a) @ (b) (c) €3 HomG(k,62 2.3, 62, see 62) have 62. Moreover, 62 Moreover, of In We also no such no also subquotient which 62 isomorphisms that (a) 36 @ Lemma €4 particular have $84. il follow will 62 141 trivial a with by l(b) We Since 62€4(61@16~) GROUP @ is 6i. has @ have 141 uniserial with uniserial The the and (641 series l(b). we N) a of quotients there z submodule. quotient filtration structure already it Hom~(142, 8 61 f we if in 62 k, is 142) Using Hom~(62, €4 the €4 is has clear 62, 6i, 142 show a @ first know by by a k. unique obtained with factors with series again 642 of €4 filtration that self-duality) the Since 62 61 Neither the that line N) F the @ Homc(k, unique copy unique 141, the 6i quotient middle factor follow M) = 62 (642 structure we €4 by with factors 0. structure @ 62, does know 160. omitting €3 M from Homc(l4p and 141 61) 62 which has no submodules no has the the of @ $ and of that 6i M) 6i quotient is known. is of (U2 84 structure top and bottom and top €4 is 62$84 8 let 62 62 in oslt extension nonsplit a 142 €4 6i 8 N €4 @ HomG(62, the 61): 62, from 2.3, Lemma 2.3, from N 62, be 2 by socle does M) 84 Let has of 62@(k$141), the we the 62 = M no uniserial 62 142 see €3 M) copy 0, be the be trivial trivial @ 62 from or that not = 14] a11 of k 0 Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 62 We shows summand has the has summand statement to statement this summand this series) can series) known, 36, Homc(62 Homc(84,36) this, are atr. Next, factors. modules some composable 14~ niques 61 (c). 4528 since complementary Lemma will summand the block. Weyl in we opt. They compute. Let The The @ @ [Ill. (a) need 8 U, recall 141 representation theory representation bc The 141) 141 using fundamental 61 Homc(641 modules first consider us first it component established This has proof block The which 141 $36 the Data l(b)) r the are @ has main @ 2 rm 2.1 from is 641 $36. 142, ~xtb(141,61) radical e easily be rest 62 summand Homc(641 restrictions and submodule are of a 642 filtration a such 141 we = we 8 CB or of statement summand by the of summands. can also can been given been as 8142 8 0, @ 61 141, aent discussed not have U. have 62 From tensor the structure by claimed in series as (a) results 61. 142,62) follows key we respectively. and @ deduced proving the 14* weights having and the the Z 62 by see @ fact the structure the of structure 2.3, Lemma 2.3, ) be the products with 160@36 principal is = 62, of 63 of Z Lemma about summand simple structure. that (P(642) in found = immediate from immediate First, 0 projective 141 Homc(84, ~xtL(36,84) U algebraic 84) the of 62 from (2.3, [15]. 0. factors main @ (the Weyl the corresponds under corresponds as = induced isomorphism The 142. modules and of Thus, in these l(b), of block 0, a omitted): Lemma 2.2, Lemma 2.2, top position part projective yet. 151 outside simple simple modules simple last just from characters, from just l(b) 61 modules, of It groups. These theorems can be found be can theorems These groups. 160, covers of Since 84) (for modules two is 61 U has modules of First, = statement for that 641, easy and 63 2(b)). weights), (c). 0 quotient the the 61 radical of k. a filtration a Gz(k) 2 will 141 U8142 modules 6. Since 160. opsto utpiiis in multiplicities composition the modules 8 the This Homc(62 to 141 we 641 isomorphisms principal 4(b), have @ 36 and make ih the With see from see second are fact the $U. will @ of layers proves and, is are E [17] the ., Lemma 2.3, 142 good (b) done. such isomorphism in (61@36)$642 allow us in that has use with The straightforward same @ and furthermore, block. the 141 now r the of H0m~(64~,62 we (b). 6? characters filtrations from filtrations of a structure a various last 62 160 factors in non-principal non-principal [6]. see unique (2.1 @ several summand follows composition @ to apply The the radical the 141, statement $ Secondly, that 62 2(i) and 36 simple in 61 above 8 of 141 inde- tech- from of that that and this and 2.3, ) SIN the 141 the @ of to f& U Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 HALL-JANKO calculations. PROOF: we from PROOF: shows by Next using Lemma using of Then of it again submodule 1. Now follows (b) (d) (c) N a By have Lemma nonzero in the since by by that P(641) P(642) P(160) (4) HomG(P the an For From Lemma Lemma factors that and induced rad 2(b). of top (a) G-homomorphism GROUP P@142 @ €3 @ 2(b). Lemma P/ the factor 142 142 142 we @ 3(b), Finally, to 2(b). rad2 filtration filtration 142, g S 36. refer which P(36) P(62) 3(b), (P we P(14, P We ~om~(rad~ 36) Since Z @ to see also need also is E 160, rad2 142)/((rad we of of $ $ [15]. $ that the k, rad2 P (~(642))~ P(84) P@ see (P(36))2 641 we by P kernel := dimk Parts (b), Parts c that 142. P P have Lemma @ P(641): radP $ €3 is 142 P) ~xtk(84,36) (P(160))3. of We radP 142, $ quotient a $ @ a + (P(160))~. C (P(160))~ now nonzero 3(b). 142) 36) (c) 84. @ P 142 = consider and Now Denote 0. 641 C of G-homomorphism = (d) N from 8 P(642), dimk G-homomorphisms and 142 by are (I), HomG(N, N is ut character just that the that the Lemma the (2) kernel maximal and (3) and image to 36) 3(c) (1) 84. A' - Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 show fundamental dominant fundamental we Now 4530 rad(641 rad(618 Considering and It for isomorphic L(wz) can module, socle V(wl) Since composition factor composition L(X) resentations nue modules induced Our the has clear of on oue ol trivially only modules simple quotient simple factors L(2wl)@ (7)). weight V(2~1 The The Since now the egt. Moreover, weights. arrive G algebraic an ascending filtration an do this, Homc(61 eventual aim eventual L(X). Hom~(Ii/rad(641 are that from V unless factors has and + composition follows data. as module C3 V(wl is self-dual, is 160). ih simple with WZ), (L(wl) ~aramentrized dimk at 142) subquotients: V radical L(wl the structure the to the In these the we [ll, stated has group G group For v(2~2), appear in appear + C3 the Since in ~xtk(84,36) that 6z @ need inequality wz 62 L(wl + 11, a filtration a HO(X) (5). is V(w1) notation L(ul k, corresponding two @ wz) ) in 8 otherwise. 4.191 to factors V 6i while = there quotient to 641,36) (61 V(w1) Then + filtrations V(3~1), = show the 8 = + [ll, are HO(wl weights (ascending, 8 recall descending order with order descending by @ wz). L(w1) wz)). 160 states @ Gz(k) of multiplicities V(wz) Tr(wl is 142),36) of respectively y Weyl by non-negative integral combinations integral non-negative 11, of by 641) that the that 5 V(w1). 8 7 is also [6], + That each L(X) dimk some V(w1 A 4.131 Until Lemma V(WI a with @ that wl Hom~(62 (see ~2 + induced and the and structures quotient fundamental result and we is L(w1 the wp), ) and Homc(rad(62 Weyl and with the t also it the ideas the have = + we further modules, and But right [6 +wz), V(wl dual w2), 2(c), + 11,531). 0 the 6i, of see tensor wz. following by restriction M7eyl the modules. wz) L(wl by that module of 8 from hand the filtration +wz) SO has factors (in factors 141 V(2~1), HO(X) that notice, 6* 641, this (3), weights We has a filtration a has factors it such the @ product + respect The modules the and side uha simple a such must 61 may (62 V(wl) are both w2) denote C3 so properties: are restrictions the @ all that representation theory representation of simple of 61 of @641), 64,. V(w2). A1 on we in can occurs 36) erpae in replaced be easy be module their factors, their induced module induced C3 (6) ascending to the of 8 algebraic the c can extend can the may 6&), by a direct a be = the The two simple, V(wl szr. Before zero. is . rational to G-module by . . 0, reverse highest weights highest calculated from calculated V(X) only with usual to replace Weyl opt. By compute. c 36). structures are Weyl by submodule Lemma X + G A5 order) are: group summand. of Lemma hence also hence once w2) the the of modules ordering modules order we induced = the turn module L(wl), Ii has other Weyl V 2(b), V shall with as rep- two SIN (6) we by by by to of 2, = is a a Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 HALL-JANKO so modules, - V modules which modules ness image ~xt&(~(2wz), only composition only itain can filtration L(2w2) extension in a Vl module for V, L(3wl). has restriction. on factor has factor by which we Lemma are Ext&(~(wz), nonzero Ext&(~(wz), strictions occurs (i) and (i) oslt extension nonsplit By Then First -+ From Wl (ii) @p2 then (i) twisting have at HomG(M the of A5/A4 the = most one-dimensional, most Every L(2wl tension nonsplit The of restricts as in L(2wl same we we now L(2wl) the 3 i) remain (ii) Thus, radG 1 restriction Z we V2 calculation of a V/Wl, L(wl) (b), is observe have unique L(2wl subquotient the dual first Z (ii) G-composition find in on L(2wl) ~(2~1)) structure + F-factor GROUP of @ Vl now L(w1)) + of HO the do so extension w2) represents a represents V/Wl. to @I Since 61,61) we course w2) a extension has either as that one (2w1 factors +wz) not split on restriction. split not it filtration L(2wz) filtration e further be nonzero a F-factor that will appearing a homomorphic a to valid no is enough is @ D M* as of = sees --+ the with + as G. L(2wz)), 2 also quotients of k @radG 0, consider only G-modules only consider a composition factor composition a Then G-extensions of all w2) of of of Ext&(~(wz), its or Homc(M, by has L(2wz) for that groups of L(w1) By G-composition the factors Wl L(w1) L(2w2) nonzero class nonzero in V filtration a 3 L(3wl) with restriction and this and it (i). [6], class G, refined by in which simple a can V(2wz). suffices 2.3, V2 to first C so The we L(wl). the @I L(2wl namely, the W2 by of find as maps be L(w1) L(w1). in by Lemma by it image and image by 61 see G-extensions extend factor L(2wl a second extension in second extension factors to c uniqueness [6], constructed. ~(2~1) ~xt&(L(w~),L(%l) L(2wl suffices to a @ 61 L(wl) Frobenius composition Since of that V2 G-submodule There + satisfy onto class in class find 61) @ and It and a and that G-modules. factors wz) This C 141, ~xt&(~(2w~), + Vz/W2 is l(a), L(wl) through if = + HomG(radG V3 in nonsplit L(2wl @ D. wz), easy is to as with W2 wz). 0, the given proves (i), L(2wz)) . Inside V. universal C a again. Ext&(~(wZ), of It and such find a composition factor composition a 6I unique by from to between V4 is as G-modules as as extensions +wz) using the follows factor the or the in see V/Wl, 2.3 Lemma 2.3 C eurd in required tensoring with tensoring G-extensions L(2wl) a the nnpi extension nonsplit a We 84 2.3, unique composition unique V5 (8). opsto factor composition is injective is preimage that (i) ls in class G-homomorphism L(2wl)@L(u2)) of V(2wz), property the of claim. @ is = H0(2wl claim that simple can be obtained Lemma L(wl) by Let 61 L(2wz)). nonsplit a V in V L(wl)@ fact ~(2w~)) and do the @ radG VZ l(a). that the are i) This (ii). 64 L(wl)) which no which @ o split not of that + the modules of of unique- l(b), L(w1 by be the be mz L(wl), are w2) k L(w2) 72 latter Weyl these both with 4531 map The is The and 2.3, the ex- re- in so E = is + a Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 principal if of of group. to now be Now observe G2 since 62 PROOF: 84, REMARK. wz), erty mation Homc(62 2(b). show 4532 does not does the calculations of by 2(a). before, 2(b). 62. we Lemma M the k. E Lemma 861 Finally, let Finally, 61 62 (k). 142 isomorphism. (b) realized (a) So @ 36. see so conditions Then head follows is that and 62 HomG(62 62 @ by that Thus by E Also, by that First, (d (e any = Thus, we we @ 62 that @ It is @ the latter k; the By have Lemma 4, Lemma 84. 36. block =) =) that the @ E of 61 Y for In 2(b) 62 proved. @ see can is nonsplit (62 by M from 61 Homc(E 2.3, dimr, has dim& 8 of any the 2 the let Then 141, = By this not each tensor E a that @62@141, @ @ the @ 61 write 61 are E shows (M Home-spaces homomorphic can E be Lemma a descending a 61 nonzero Lemma proof 62 641,84) principal (a). @ nonsplit We for Ext&(36,6]) is H' hard and of tensor product tensor be the Homc(E@62,36) €3 we hce o all for checked €3 a has 62 extension a e obtained be rdc with product (G, E the 641 is nonsplit extension nonsplit 62) then €3 have oslt extension nonsplit a that oslt extension nonsplit have right contains of 84 as to the a 62,84) 84) 2(a), )/ 2(a), three @ 61) map Lemma submodule M are extension show rad Lemma block Ext&(36,61) 61. the Hom~(6z restriction Homc(6z hand = @ we image shows filtration the in the of special 62 1. = Since Hom~(84,36) conclusion 61, 2 the Hom~(62 that have @ 36 by 36 summand 4, iial fo etnin between extensions from similarly 1. side first Homc(E, ny composition only of 62 three where 61 2(b) unique we is which Hornc(k, Y that r 62 has @ @ @ 3-factors of all 0 factor of the saw which Homc(E, 61, with 641 6&, # 62 with # of will (6) special the of of M 36 HomG(84, restriction 4 is 84 Ext&(141,62) @62 of is 36 is composition @ 0, 36 then that 1 61 will of 62 61 the is factors has as imply corresponding k = 14i,E) so nqennpi extension nonsplit the unique 84 is by M by 8 @ €3 by a a composition a 0. or E be oslt extension nonsplit a the the it we $84. types oslt extensions nonsplit a nonsplit @ 61 62 141, 6i. (62 @ 36 61 in This submodule isomorphic a is shown 62) the @ hl aim shall @ nonzero class nonzero 61 62 nonsplit 62 only @62)@61) atr with factors of (i) or a and As E) 2 142) Combining 141) of @E) @ @ homomorphic image homomorphic one completes factor uniqueness a and HomG(84,62 F must we 36 Homc(M, 641, nonsplit extension nonsplit 3-factors and extension this submodule this to simple module simple H1(G,84), g for = Weyl have # E (ii), be extension k2, to 0, 36. be surjective, be atr Next, factor. 0. €i2 and the zero once zero = that by prove by Finally @ mentioned module in (a) the this 0, the Therefore algebraic of we 62 of 62) Lemma Lemma Lemma of simple in E proof. @ so E prop- infor- @ 62 of have that k @ = can E). the SIN (b) 6,, the we up we by by to 84 62 of in of 0, Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 modules for modules HALL-JANKO with the block aia defect maximal for tic are Our n te character-theoretic other and tables 572 have of where more where there.are nue from induced same purely module, in decomposition the By of modules simple the indecomposable are character of "1" Q@ and We The In This We Before our calculations), their rad rad hm eut from result them 2. tensoring the some differences processes uniserial blocks and extensions. are S. notation hence simple modules 61, 53 method yields method label has will Q Q, in simple modules in simple modules character-theoretic completes In 36-dimensional ones. 36-dimensional conjugate @ 133. we we 62, [9], going a a practice, table eight S, of give we the common can routinely can sue k assume few the complicated 14, projective indecomposable projective the defect which All [12]/[13] follows GROUP of any filtration any obtain of simple modules details into The and 361, algebraic simple modules, simple of we restriction summands. more projective indecomposable projective these the to the g. if must be must obtain the 1 in labelling in details, decomposition structures situation that 362, those a 18g1, S are calculation and information stubborn to tensor are of tagtowr application straightforward Besides is small (we is shall extensions in arguments easily be the find group may 501, of the self-dual. question a projective module projective a [4]/[16] with 18g2 and isomorphic an we From [9], projective calculations only for only calculations rdc into product principal the 502 information be in blocks in of hs, there these, found by twisting algebraically sketch of having 4-restricted highest having and where decomposition where cases "2" of as very decompositions the this, and ordinary their (three about the about matrix. simple characteris- module in extensions whose (but by does In 216, the from projective indecomposable modules projective few ((236,236), 236. the the we the block to modules in dimension. of by other use modules use still of and itnt labellings). distinct the summands a derive solution defect other the a be may method the outer automorphism. outer Gal(F4/F2). modules in modules indecomposable and submodule are three are All For closed are nqe non-principal unique three indecomposable block Brauer cyclic on of 3-modular the example, with block information 1. k, P@ these (236,501) which the of field can we found. blocks The of same kind) are same kind) Then 131, of of matrices, Brauer lc. Moreover, block. S P the blocks trees maximal defect; maximal most of the principal shall of be Q@S and dimensions 6 modules with modules the are of maximal U2, we method the suppose if of cyclic read characters weights, Note characteristic (see projective self-dual use Q summands S and have about the of 1. head in The 211, @ summands is off eet zero, defect [7, block. to the S blocks that any simple any (501,502)) filtrations defect and into principal 212, P of from compute VII.121). block needed. through block except and 13 and and P in suffix there cover trees head their then 4533 with @ 571, The is the the the we 3. of S, in of of Q a Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 4534 using of interest LEMMA 3.1 block. between The trivial is of PROPOSITION LEMMA theoretic avoid written defect LEMMA The zero the principal (d) (d) (b) (d) (b) (a) (b) (a) (a) (b) (e) (a) (c) (c) (c) (f) The first this information, irrelevant composition cases: 36 90 P(1261) ~xt&(l3~, E~t&(571,13~) P(36) ~xtk(571, Extk(212, P(1262) Extk(212, P(90) 1, E~t&(57~, 126, 1262 P(36) P(1261) and the and next in and 2. simple modules in modules simple 3. 1. is with principal @ GJ information. the exponents. the obvious, @61 @ 131 block We many extensions many We he lemmas three @ 63 : 61 composition series composition 36,9O, 4. 63 63 third 131 131 : information, zo have ro have go 1261, 572) k) k) 132) 131) 61 61 components. The factors {k2, (k2, So go = = 3-block. go the {1312,2122,5712,133). {k3, go is 36; the the = = = = 1262, 0. 0. following in P(132). P(131). because The P(212) P(133). 1314, i31,~32\2i1,5722, second is second 0. 0. 0. 0. 1312, conjunction following following of simply the 126]; computations we P(90) tensor 132,212, of 1323,21i2,212,572, In 83 principal 211 simple will hold: this as P(571) because collect uniserial @ : usually write usually products opsto factors, composition given: 90,36,9O. subsection, with 212 5712, P(1262) modules lc. Let block. has no together some relevant some together the involved the 133~). 133'). projective modules projective of corresponding Cartan : can be can 126~, simple modules. simple component in component we "Zen 1334). are routine. are us compute 1261,1262. matrix and matrix deduced to is dispose first mean isomorphism with Cartan the multiplicities the immediately in In knowledge extensions character- principal blocks order to order of number some SIN of Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 that PROOF: HALL-JANKO lemma which to modules and Parts LEMMA To (c) (b) 133. (a) the Lemma go (a)-(e) is 36 1261 the It isomorphism 5. easily further, This S E follows socle series is Z 6i 2(b) now GROUP a ~xth(212 uses only uses 8 verified composition 8 62. we implies that follow 212. need for are described. (using self-duality) the parts $571, from 61 that any to @ factor look the 61 rad simple module simple S) fact E P(l261) (a) at 131 of Hom2(rad(1261 k k and some that rad(1261 (b) 8 from none small 6] @ of S 211 has 8 other character Lemmas of modules. 61), a the @ simple head isomorphic simple head than 61), and corresponding simple corresponding 2 data. S). and 133, (f) In the from 3. we Note first Note following have (a), 4535 (2) (c) Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 be a be no h symmetric the into PROOF: qae has square only 212@131 proved. (b). the P(132), and establish PROPOSITION (g) (h) map 4. PROOF: (d) (b) (h) (a) (g) (k) (c) (e) By (f) (j) (i) (1) holds. follows fact Lemma direct the Parts in self-duality Extk(571, Ext;(132,211) ~xtk(212, ~xtk(132,57~) ~xtk(l32, Extk(ls2, ~xt&(l3~, ~xtk(133,572) Ext;(133,133) Extk(133, Extk(133,211) Extk(211,211) the That By that module The To (i). It summand as its from (c) right one-dimensional Lemma 2(d) {U12, prove is it composition factors composition the 6. and easily is this. structure that hand 57, k) 571) module in(f) 133) 132) k) a tensor a we 57,'; (a) (d) 2(d), Z = and ) checked Similar considerations Similar 628 have ======S g = k. 0. side and follow 133). 0. 0. k. 0. 0. 0. 0. k. k. 0. alternating so 14 needs dimension factor ~oc(l26~ of (b), (f) has fixed (2) has the that from Since must holds we proof. simple of are points. 16' return the [131@131 @J 36 have simple is by squares band from obtained 8 6,) not €3 There head decomposition self-duality. It self-duality. 13, simple 131, 2 to divisible For of : 212 and (2) : 501] and is which spaces 2121 (e), nothing submodule $ head socle in = the 571. = by the 0 we in the of 0 and = 132 3 fact of then andsince turn We proof kG-homomorphisms calculate to [3, [62 character A the socle so, prove Prop. claim that 571 €3 follows is by of tensor 211 a quotient a is in self-duality, Proposition 133 the in clear that 2.21, first its : data, 501) from (a) cannot square tensor kernel. (e) every from that and SIN and (f) so of is Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 5(f) We by need U. homomorphisms the to into HALL-JANKO factor follow factor phism results mation, epciey Since respectively. structures this HomG(131 By obtain (i) information, using information, that P(133) in Now 14, By with factors with factors Next Lemma (g) and (g) 131. follows We Lemma 501. Lemma claim module and has ~xt~(l32,131 for to examine to 36 from of is (3) for compute: (Proposition 4 (Proposition 62 we 20 Since we any simple any @ (i), itain ih factors with filtration a and 131. 61 that 8 in from S of consider 2(d), 8 P(1262 (h) 131 5 can 5(b), the 61 8 equal any nonzero any 14, $ the 131,571) the (b) GROUP the [I31 61 The @ U of 212, 2 deduce claim (h) 62 which proves proof to ~xtk(133, 1262 we 211 factors are given is ~xtb(132, we whose that quotient to @ 62 8 (3). @ @ the the claim a quotient a module and 133, 571, have 571 the 61). have has 501, 571) @ @ and (a) g of lemma module We (g) latter has latter (61 61 then so it principal tensor k, : homomorphism from homomorphism Proposition 4 Proposition a and From 1261 now 62 1321 (2), 1262 of can and 2 so uniserial S) @ S) S @ ~xtb(132 62 is (j) rad(62 (5). part $ 131 131) and of 8 2 using Lemma using follows = P(132), E now 14. (h) @ product its enough @ follows 131, 6, Hom~(rad(l262 61 0, Hom~(rad(36 61 simple head simple @ block 62 It in Lemma in (61 filtration from (c), (c) The has, S obtain we @ 571 @ remains submodule 8 we 212 @ 131. of rm Lemma from @ 61 which implies obtain (d) so1, 62 to parts structures from has 211) have with (4). by this proposition). this 131,571). 8 @ @ prove (1) Assuming (e) and (e) (61 Lemma 211) a 62 induced (211 3. Next, to more 5 by of from Lemma from (i). 212 unique 61 an U1, that (g) 8 @ @ verify 368 Lemma U @ by @ @ 61). embedding To (14 131), has of and with closely. we 131 y Lemma by (f) 61) 61), Since H0m~(13~, 5(e). 62 5 131 prove by these submodule this, $ the The 571 note (f), o follow now S). to (h). @ and S). 62), that 2 composition and rad 61 U claim. (d) 131 we as only composition only Now 5(d) By Now factors (k) filtration a that From by @ has no has 62 of ~xtb(13~, has a @ obtain 21 Lemma using 2 Lemma 131 @ composition P(1262), and 132 we (a) P(36) the By isomorphic (c), a filtration maps onto this XI1. from are @ examine nonzero and existing go Lemma isomor- (4), factors 5i1) we infor- 8 given 5 with 4537 5(i), 131) The 133, this 131 (b) (i), see (3) (4) we we % Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 The Extk(k, Table 4538 but Brauer by 4.1. two simple modules simple two simple modules taining following Up u oain matches notation Our In The The f4. these play these fact $4 The 2. to group projective indecomposable projective characters, There the we 131) that automorphisms simple structures: assume simple modules simple S G this are no k has 70 kG-modules (part part space also decomposition and k five of 4.la) to in our in dimension two (c) 90, 5-blocks. be is and [Q], of not an and The uuly dual mutually 175, this proposition). this hc is which will discussion. duality lerial closed algebraically zero the principal 225 be numbers 14 There oue in modules follows principal denoted of and we our which one is one which are 5-block have 300, simple an source main from and products of and three block, by the now a their dimensions. their Lemma 5 Lemma block of field modules block blocks with computed G. faithful; this faithful; of of for of Cartan of characteristic defect (c) defect of data characters. defect zero con- zero defect dimension and all matrix concerning 1 1 we entries There are (j), with have denote using 5. two SIN the 50, in Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 multiplicities n braces in whose containing HALL-JANKO (m) We The (p) (q) (k) (h) (n) (0) (g) (d) (b) (s) (a) (e) (r) (c) (f) (j) (1) (i) now 6 686 6 6 6 f4~3 6 6 6 21 21 f4 14 14C341 Cartan 14 14 14 14 14 group 8 8 C3 8 8 8 8 8 8 8 C3 8 8 8 8 8 indicates a module having module a indicates 4.lb) 90 41 64 56 70 85 f4 501, describe some describe 90 70 41 21 70 90 6 21 189 14 Z "= Z E g 2 g Z are E 6 GROUP = matrix E85$189$300. Z g Z k g 502 Z (6, 56 21 {6,56, The (6, 14 6 14 has $14@21. {1+4~,56,643,2023) k {64, (14, {k, k {143, 14 indicated $ {14~, $70. $190. $85 $41 $21 $14 $14 562, f42, and $21 f4 four non-principal5-block 143, 564, 212, is given below given is 642) 21,85, $ 56,64~, 190,202). 21~,41',85~, $70 $21 $41 $21 $21 350 214,41, 64. $189 further 643, tensor 412, by $350. respectively, $90. $189. $70 18g3) 190) 8S2, an 202). $70. 85, products blocks exponent. $ $90. 18g2) $350. $ 18g3) ~~350. 85 has 18Q7} {702, the $ containing faithful modules. faithful containing $175. maximal are 189 $70 of nlsd composition enclosed of $ go2} simple 70 of $90. @ of defect zero, while zero, defect $90 $225. 90. defect. maximal modules. $175 $ defect. 225 An factors $300. expression the Three, fourth 4539 and Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 position 4540 modules. LEMMA PROOF: er as pear calculations. LEMMA (maximal semisimple quotients) semisimple (maximal we occurence since, for the PROOF: consequence: 90 are then are self-duality, mension another. contradicts divides REMARK. l(a) Next, From (b) (a) (d) (h) (c) (g) (d) (b) (e) (a) (f) (c) $ (e) (f) (i) need structure and 90 would 90 P(90) P(70) P(90) P(70) P(90) P(k) P(70) P(k) P(k) head(70 head(70 head(90 head(9O head(70 head(90 we subquotients the the 2. and 3. These of Upper only We example, easily Again, that of A consider every direct what @ @ @ dimension above two lemmas, two above etc, self-duality. @ @ @ @ @ @ general result shall use By 225 of Let 21 6 14 determine 6 of 6 results @ @ @ 14 8 @ @ 14 be f4 f4 found bounds a self-duality, the P(90) E " r using is clear is 6) 6) we 14) 14) f4) f4) E r M 2 Z 189 P(6). a 90814 P(64) ~(f4) P(14) P(21) 2 tensor S P(14) P(14) direct ~(f4) proof is proof P(6) know F E Z in filtrations in be this from by all 64 f4 of that on 6 Lemma 14 14 f4 summand the from $ $64. $56 the follow an $ $ lmnay considerations elementary $ is a $ $ here @ $21 $64 the $ $ products $ about summand [3, 350. 3502. P(64) P(56) P(85) P(70) Lemma P(90) component pure the 189 indecomposable P(189) P(64) P(21) quotient multiplicity Prop. $64. multiplicities Lemma and $189 22 $350. $202 we from 1. $70 socles of P(90). calculation of $ $ of $300. $ 8 on one later $ $ can obtain information obtain can e give We of eti tensor certain 350. $ 2.21 P(64) P(90) the l(f). projective character computations, 14 of P(202) P(189) $70 $90 of small projective P(70) l(d), are in 90 had tensor states P(90)@ of @ $90. $ a By the $175. @ are that block 225 a 90. 3504. $ $ $ the module with characters. with 14. Lemma same. subquotient that occasion. P(70) p(90)~ 3503. product given modules 14. of details It If of products if would it 14 The of by then it then maximal $ the $225. modules appeared follows l(e), composition factors, composition of P(90)' the in subsequent in exact for It field that then has about {905, previous 70 which (a) divides block characteristic from multiplicities defect the $225. module with with simple with appears, follow only. twice 701, the following will decom- Lemma lemma the of heads which ~xt' from then The SIN ap- one di- so Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 where see have composition have of HALL-JANKO later extensions between 4.2 Here by is rad of tient computed from formal computed of process. of ideas. Lemmas referred Lemmas ideas. succeeding PROOF: PROPOSITION k nor k (m) In a 2 a the projective (h) extensions (b) (k) (d) (g) (a) Ext' (e) (c) certain filtrations (f) (1) (i) (j) that P(90) Extensions. application. the of we on 41 modules Extb(6,56) ~xtb(14,21) Extk(l4, ~xtb(6, Extb(21,k) ~xtk(14,14) Extk(21,85) ~xtb(6,64) Exth(l4,85) Exth(189,85) Extk(189,41) Exth(21,41) Exth(6,6) P(70) k groups The the We could also have also could appears. the is result first and is can be can propositions denotes isomorphism denotes corresponding simple modules, simple corresponding leading diagonal, it diagonal, leading quotient a modules rest require rest consider GROUP for @ 41 1. f4) whose k) between 14, hrceitc 5 characteristic = are not multiplicities filled in filled By In several In ======k. whose the of = = k. 0. k. 0. k. Lemma ih simple modules. with 0. 0. k. k. k. heads manipulations the 0. k. P(90) to we of filtrations simple modules. simple some rud sn eaie projectivity, relative using argued quotients immediately, by filtration head apply P(70), instances, are number 2, and extra [90 is we is clear is given the will of P(70) given it @ of of have radP(9O) parts arguments, with same method follows 14 rad(P(9O) be are P(90) in Lemma in where and : that in will found k] from the in Further Lemma of = so ery all Nearly @ that be there are there the the equations course, 0 @ 4.1. these entries these 14 with @ used but and principal 14 3. Cartan radP(9O) and 14), to 3. extensions based on still based Then C little [90 to other ~xth(k, In particular, In only proving P(70) entries in entries P(90) of calculate a calculate @ number is number Lemma difficulty soc lc. Since block. 14 tensor split extensions split but @ in @ : k) M 8 can 14 (a) the table the 411 14 o in not = 14. the C is the products I. and = then induced 0. number by zero neither rad a In tables 0, Since same quo- 454 (b). this our the M. we we be or of 1 Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 4542 is k (a), (b) Since (P(70) and We but contrary rad Lemma rad It latter. Thus, by with 85, PROPOSITION 2 parts (g) and a Now Let Now (a), (b) Next, Parts is Next (b) (a) (c) self-duality, claim quotient then / / and similarly for similarly clear socP(189) soc(P(14) we 41 already proved. already us Extk(l4 @I ~xt;(k,41) ~xtk(21,21) ~xtk(6,190) consider This (e) and we 2(b) [rad returning using self-duality, using (i)-(m) already to return do 14) the (c) from that is consider (3). the not proves of $ / and that immediate from submodule soc(70 2. (P(90) P(70) the $ rad self-duality @ ~xtb(14 the to has ~xtk(21, Cartan appear. are know 1 Now, 41. (d) rad 189,85) 85 P(189)) to the = image / (f) filtration the = = P(k) hold. easily soc(70 8 is the / $ 0. 8 filtration 0. k. Ext?-(14, soc(9O and not 14) P(90), matrix, same 14), $ By filtration being in being k) @ by % G If we has derived 21 : a that (g). @ 14 851 Z Lemma not, then k. projection whose quotient rad property. @ see Lemma $ 14) Homc(rad it and a From it = of k 14) 189,85) follows P(7O) unique that $41) go follows of 1 imply P(90) the from P(k) head Go 2, P(70) {212, (5), ExtL(21, rad l(a). of kernel 14 Since = 8 onto that @ subnlodule @ the rad is / that / 0 $21 14 (1) k, (h). 14 21. @ soc(70 soc(708 given Homg(rad/soc(70 and P(90) exactly 41,14,189,85). c identities 14, rad IP(14) of and at the 9 P(70) k) Extk(189, a we P(90) 85 in &3 = self-duality quotient @I 14) see Lemma @ 14), beginning. 85 one 0 : 14 @ 851 and 189. in has @ from and 14. so k), of = 14 Lemma Extk(21,41) k we a rad map. 1, it Since 3. $41) unique quotient unique is a is quotient (2) it have @ In particular In / We know by know We follows must be socP(14) and quotient 14),85), rad = 1 Therefore by and 0, Lemma k@41, P(7O) from (I), = that SIN (3) the the (5) of or 0, so Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 REMARK. are from Lemma from and and Ext$(21,189) this Lemma which gives which PROOF: HALL-JANKO any simple any 14 189,21) Lemma rm Lemma From To (c) (d) (e) (f) and known. now all extensions all dimension is dimension prove follows Extk(21,189) Extk(189,189) ExtB(14,189) the structure the = 1 If (d) to It l(a) dimHomg(rad(P(90) quotient ExtL(k,41) (f), a reduction a is obtain follows from Lemma from l(g) GROUP # now and we 0. 2(h), at and in use Lemma use Equation easy (b) most from existing from of = = the of = Proposition 41 yield Proposition k. k. to # 90 k. to first 2. 8 rpsto 1 Proposition 0, l(h) @ see 14 Now (1) then 14 and that l(b), in using @ and of (e) results. the last 14), by l(f). rpsto 1 Proposition P(k) l(c) (a) 70 follows (a) principal Lemma 21) terms @ and For and and and has and 14 described (b), from (h), other (a). Proposition radical Lemma from 2(g), block. we to shows Proposition than the filtration the obtain use 14 (and Therefore l(h), extends nontrivially extends hr, one there, that parts the 1. socle) series socle) one in one it dimExt&(l4 (n) l(i). follows of (a) and sees question P(90) follows (j) that that of €3 $ Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 4544 and PROOF: Since filter quotient (b). (c) do We it head Lemma nonsplit quotient must (c). subquotient Next (rn) (b) (d) follows by follows (a) (h) (g) (k) (n) (c) (e) (f) not (j) (i) (1) (using-(a)) have 90 Also Since of P(90) be these modules these Extk(64,202) ~xtb(l'4,202) ~xtb(f4,56) ~xt&(f4,64) Extk(64,56) ~xtb(202,202) Extk(202,190) Exth(202,56) Extb(f4, Exth(56,190) ~xtb(64,64) Extk(85,41) Extk(56,56) ~xth(41,41) extend @ we rad l(s) semisimple extension From of 14 6 Homg(6 since consider they have P(so)@~~, P(70) 8 and and of (b) f4 f4, Lemma and 6 [rad f4) also filter also we the that Z Lemma @ which @ @ of ~(f4) the = have = = = = (d) 85 = = = / by 14 following = = filter = = 85,56) soc(90 56 dim~xt$(202,56) = = 0. k. k. 0. k. 0. k. k. we obtained 0. k. k. and k. self-duality. Since self-duality. 3 tensor (using yields 0. 0. by seen 3(e), (c), $ see P(70) P(90) since 202. P(64) @ 2 that that the (d) structures: (e). products (b)). 1^4) [rad Hom2(85, by @ @ By [90 and @ dirnExt~(f4$64$202,56) Next : 6 6 / omitting 561 P(202) The Lemma soc(9O @ Z E Lemma composition factors f4 ~(c4) P(64) = we of < module any direct any 6 : 1 P(70) 1. consider 1901 @ @ $ and l(o), the To prove 3503. $350' f4) $ 56) 1 = P(64) 56 (p), and rad top and top : 6 Z 0, 2021 @ 90@1*4 appears Since summand we k P(9O) (q) we P(90) 85 $350, (h), by = deduce can deduce can we rad 1, bottom Lemma 6 and {6,56', we @ only once only with and (g) see 5 P(9O) we 6 of need 70@f4, has 2 (f). follows that 56 6@ also from 6 factors 202,190). l(1). and @ of 90 of a to Since 85 must (a) 70 obtain simple f4 which in which find a find from f4. SIN and €3 The is @ the by 56 6 6 a Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 HALL-JANKO not one not ae ieso divisible dimension have since P(6) and hand we Since and this and has By PROPOSITION 0 factors mod factors. and From block tion PIP2 dimHomg(rad(P(70) would To prove To By From We Finally, have our rpsto 1 Proposition all PI $ 5 Extk(6 we rad Lemma of k prove are both are the 2 S' of P(56) extensions claim, @ and and the Now contradict C maximal have Ext Ext&(6 / the Extb(41,41) we structure SOC(~O@ Pz (i), filtration so (I), &(56,56 the dimension $64,190) @ (8) complex GROUP turn Ext&(85,41) l(c), the 4. C we we P(64). isomorphic (m) @ implies composition P defect is defect ~xt$(85,85) (1). dimension except have found have to 56,4l) note (a) (7), f4) where and of $ of the @ Now character and $ Neither E 190) so at = P(70), that P(90) f4), (I), by (n) ~xt8(85,41) last {6', Extk(64,64) 0 2 most (i) P(6) PI # using to (j), (by Proposition 5 1 ~xti(56,6 56) (m) together. the part the of holds. 0, 562, missing factors is the 70 (Remark @ k = we 90 we for the $ 2. of @ quotient a <: f4 and Lemma 1 Lemma @ 0. @ 64,190) P(56) desired extension. desired have Then Extk(56,56). ere 126 degree see example from 3. f4. last we f4 (n). entry. of of r are First obtain nor Since that @ fe Lemma after Consideration member $ P(70) Ext&(21$41$85 (c) and (e) (c) and rad 41) P(64), already , (1) 70 the of 2(b) we P / the @ and soc(70 has P(90) @ assumption = claim f4 is and part the f4 our 41 (n) known, whence known, P(70) has at imply in fact @ @ and the 3), claim and dirnExtk(56,56) most of of the 190 f4 f4) we the @ P(70) Cartan that 85 known (j). while relevant as Extb(56,190) follows 2. $189,41) f4 shows see as heads a the On has composition quotient, so quotient, @ that P2/P1 matrix), extensions reduction f4 the (k). from ht the that filtra- a of block in 202 other these 4545 <_ and the (b) (8) = is is 1. Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 4546 block 1 We follows The PROOF: soc k Since from is factor is Since images Now HomE(F3, and of $ self-dual not r the 4. 2. 6. 5. 3. 1. P(85), will 14 lemma all J. P(21) for Soc. ematics, Lecture J. D. finite groups," Clarendon Press, Oxford, Clarendon finite groups," appear, consider M. M. (1986), Benson, D. the structure soc [6@85 part simple, second of L. $ that H. prove of the Benson, Representations R Dowd, Ignoring P(85) 32 Alperin, this filtration this 12 Conway, Bremner, so F, 85) of $ and will 329-357. composition Notes Communications (1972), $ : Marcel Dekker, Marcel P(85). soc2 that in soclP(k) space 1901 any we 2 41 P. "Modular Representation Theory, Representation "Modular is J. then 21 F3/ D. Horn3(21 Sin, see the 1081, R. P(21) $ components outside components F. dimHomz(F3, nonzero = 87-88. R. of does not does Gorenstein, by 85 T. soc We follow Carlson, 1, that unique P(k), V. of On Berlin, Curtis, F3 $ 1. we 8 of Simple Moody, P(85) define C factors representattons 8. rm existing From 189. 21, Using socP(85) New obtain rad2 F3 4 in Algebra in 8 from we extend 85, extend minimum rad2 E 1984. Nilpotent 41,85) A S. i Lie which ok 1985. York, have the (i=l, J. Homc(F3, vantshnng P(21). = P. 85 belong 85 Lemma (9). 85) P(21) Patera, Algebras," our Norton, 1,. filtration - F3/Fz is and it and 24(8) Dual 5 2,3) a 85 has estimate . elements submodule .5. in the Since of 1, we $ - theorem 1985. "Tables l(i),(j) (1996). R. the while 85). leri groups algebraic then rad to to principal have Monographs Thus r remains A. Parker, A. by as New (9)) 41 kernel P(85). (41 an results and We 2597-2686. for setting of - F3/F2 ~om~(rad'(~(21), a and 8 ~om~(rad~(~(21),85) the Fs/F4 8 Trends and we Dominant of composition cohomology, 21. use the - 21) of to rad Green (n), block, have R. on in the proposition F, Thus, 4. bound $ Z A. Pure P(85) in we (k xesos it extensions, to F2 Let F3/F2 Tang, identification Wilson, Weight 21 characteristic we 821) ehd, Springer Methods," be have and Proc. Amer. G we Fi the and and the factor. have the J. soc2 Applied have shown have is $ denote Multiplicities Algebra "An dimension is proved. F4/F3 85)) principal (148 the P(21)$ P(k) Atlas latter Also, Math- Math. f # only now 2, (9). SIN the 21) (9) 104 z @ 0. 0. To of Downloaded By: [University of Aberdeen] At: 16:50 15 July 2007 HALL-JANKO GROUP HALL-JANKO 10. 11. 13. 12. 14. 17. 16. 15. 18. 7. 8. 9. Theory W. J. G. G. W. ford, in Berlin/New A. A. 1982. tions Logic P. Sin, P. R. P. ture P. 13 D. (1988), 1987. (9) Algebra Landrock, C. Landrock, "Finite Landrock, S. Parker, B. Hiss, Hiss, S. Feit, Feit, (1969), (1992), Notes, Kondratev, 1989. in Algebra in Wales, 27 Kondratev, Jantzen, 333-349. 11, On Blocks "The K. K. (1988), groups A 513-516. 16(2) the Cambridge 2653-2662. Lux, Lux, "Brauer Trees "Brauer Lux, York/Heidelberg/Tokyo, Generators collection The Representation wtth l-cohomology "Representations 6 429-444. Decomposition The (1988), and non-prmcipal Decomposztion (18) cyclic defect cyclic Group rur Characters Brauer orders, University of (1978), of 357-398. oua characters, modular the Algebras Theory of Proceedings, Hall-Janko of 2-blocks 1865-1891. the numbers groups for groups of numbers Press, Sporadic groups Algebraic of 1984, and Finite of Cambridge, of of their Ottawa, some group pp. the of G2(2n), the the Hall-Janko the rus" xod University Oxford Groups," Groups," North Holland, North Groups," the Unpublished 25-63. sporadic Groups," groups Modules," sporadic as group Lecture Communications a subgroup 1983. J2 0 simple Jz, groups, Academic Revised: Received: and London. Notes group, (1988). Algebra Aut(J2), groups, of in in Mathematics," in G2(4), Communications May "Representation Math. Press, February in and Communica- Amsterdam, Algebra Algebra 1996 Press, J. Soc. Logic London, Algebra 1996 Lec- Ox- and 27 20