Journal of Algebra 234, 492᎐506Ž. 2000 doi:10.1006rjabr.2000.8552, available online at http:rrwww.idealibrary.com on
On the Automorphism Group of Extraspecial 2-Groups
Peter Schmid
Mathematisches Institut, Uni¨ersitat¨¨ Tubingen, Auf der Morgenstelle 10, D-72076 View metadata, citation and similar papers at core.ac.ukTubingen,¨ Germany brought to you by CORE E-mail: [email protected] provided by Elsevier - Publisher Connector Communicated by Gernot Stroth
Received April 3, 2000
TO HELMUT WIELANDT ON HIS 90TH BIRTHDAY
INTRODUCTION
ކ Fix a nonsingular quadratic space Ž.U, q of dimension 2n over 2 G s q y Ž.n 1 . Then either q has Witt index nqŽq2 n. or index n 1 s y s Žq q2 n.Ž.Ž.. Let E Eq and O q be the associated extraspecial 2-group 1q2 n andŽ. full orthogonal group, respectively. Usually one writes E s 2ifq s q s 1q2 n s q q2 n and E 2y otherwise; the orthogonal group is written OŽ.q " , O2 nŽ. 2 . It is well known that the group of outer automorphisms OutŽ.E OŽ.q , acting on ErZEŽ., U in the natural way. By Griesswx 3 the group extension
U u AutŽ.E ¸ O Ž.q
is nonsplit for n G 3Ž. and split for n s 1, 2 . Indeed, it then represents the unique nontrivial cohomology class in H2 ŽŽ. O q , U .Žsee Theorem 0 below. . Although the extraspecial 2-groups and their automorphism groups have been studied extensively in the literatureŽ e.g., seewx 1, 3, 4. , some interest- ing questions remain. In this paper we study the automorphism group AutŽ.E and its projective representations. We write A s AutŽ.E or, indicating its dependence on q, A s AŽ.q . There is a parallel theory dealing with central products of extraspecial 2-groups with cyclic groups of order 4, leading to corresponding extensions of the symplectic groupŽ see Appendix. . Extraspecial p-groups for odd primes p behave quite differ- ently.
492 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. EXTRASPECIAL 2-GROUPS 493
Ž.1 Automorphisms: Recall that a group is complete provided its center is trivial and every automorphism is inner. We show that A s AŽ.q s q s q q is a complete group except when q qE26Ž.dihedral or q q ŽO26Ž. , S8.Ž. In the exceptional cases Out A.has order 2Ž Theorem 1. . The proof does not use the known fact that all automorphisms of OŽ.q are q s и inner, up to the ‘‘triality’’ automorphism admitted by O84Ž. 2 D Ž.2 2. Ž.2 Schur multipliers: From Steinbergwx 12 it is known that the Schur s ޚ s multiplier MŽŽ.. O q HO2 ŽŽ.q , .vanishes except when n 2, 3 Ž where it s q has order 2. or q q8 Ž.where it is elementary of order 4 . We prove that MŽ.A is an elementary 2-group and that the natural map MŽ.A ª MO ŽŽ..q is an epimorphism whose kernel has order 2 or 4, the latter only occurring s q for q q6 Ž.Theorem 2 . Ž.3 Canonical extension groups: The canonical isomorphism ; OŽ.q ª Out ŽE .may be regarded as a coupling of the groups OŽ.q and E s EqŽ.Žin the sense of P. Hall and H. Wielandt. . The task is to construct all group extensions E u G ¸ OŽ.q to this coupling. In general, such extensionsŽ. with nonabelian kernels need not exist. But in the present situation existence is guaranteed by a result of Griesswx 3 . We shall investigate in detail the cohomology involved. It turns out that there are exactly 2 и 0. BACKGROUND s g Let Ž.U, q , E Eq Ž.Ž.,Oq , and Irr Ž.E be as in the Introduction$ s n s ކ m s ŽŽ.1 2 . . Let Uˆ HomŽ.U, 2 be the dual space. We identify U U Uˆˆm U in the obvious way. The assignment uˆˆm ¨ ¬ u ˆˆˆˆm ¨ q ¨ m u de- fines an endomorphism of Uˆˆm U whose cokernel S2 Ž.Uˆ is the symmetric square and whose image, L, is contained in the kernel. We may identify 2 s 2 ކ g 2 S Ž.Uˆ H ŽU, 2 .via the Kunneth¨ theorem. In this manner q S Ž.Uˆ is s s 1 ކ the cohomology class associated with E EqŽ.. Embedding Uˆ H Ž.U, 2 into the second cohomology through the SteenrodŽ. or Bockstein map we get a natural exact sequence ª ª 22ކ s ª ⌳2 ª 0 Uˆ H Ž.U, 2 SUŽ.ˆ Ž.Uˆ 0. We have L , ⌳2 Ž.Uˆ . The symplectic form q˜ related to the quadratic form q should be considered as an element of L. LEMMA 0. Excluding the case where E is dihedral of order 8, OŽ.qis absolutely irreducible on U. Then q˜ and q are the unique nontri¨ial elements in ⌳2 Ž.Uˆ and S2 Ž.Uˆ , respecti¨ely, which are fixed by OŽ.q . For the first statement see Proposition 2.10.6 inwx 6 . Since Uˆ, U as an ކ g 2OŽ.q -module Ž self-duality . , this yields that q˜ L is the unique nontriv- ial OŽ.q -invariant bilinear form on U. Now q is OŽ.q -invariant. Thus we mayŽ. and do identify q˜ with the image of q in the above exact sequence. EXTRASPECIAL 2-GROUPS 495 Since OŽ.q does not stabilize any nonzero vector in Uˆ, we get the uniqueness statement for q. We mention that q also is the unique nontrivial element in S2 Ž.Uˆ fixed ⍀ G ⍀q by Ž.q provided n 2. Sometimes it is useful to remember that 2 nŽ.2 s ⍀y s2 G ⍀y s DnŽ.2 and 2 nnŽ.2 D Ž.2 are of Lie type Ž.n 4 , as well as 4 Ž.2 ⍀q s ⍀y s2 A16Ž.4, Ž.2 A 3 Ž.2 , and 6Ž.2 A 3Ž.2. There is also a natural isomorphism MŽ.U s ⌳2 Ž.ŽU Kunneth¨ . . There- fore the above exact sequence may also be considered to come from the Universal Coefficient theoremŽ. UC . Recall that if C is a trivial A-module for some group A, there is a natural exact sequence Ž.UC 0 ª Ext ŽArAЈ, C .ª H 2 ŽA, C .ª HomŽ. M Ž.A , C ª 0. Whenever C u G ¸ A is a central group extension, its cohomology class ª is mapped onto theŽ. transgression homomorphism G:M Ž.ŽA C fitting into the exact homology sequence MŽ.G ª M Ž.A ª C ª GrGЈ ª ArAЈ ª 1 induced. . A projective ރ-representation of A can be lifted to an ordinary representation of G if and only if its cohomology class is contained in the image of the dual mapŽ. Schur . Here we identify H 2 Ž.A, ރ* with the character group of MŽ.A via Ž UC . . Now let A s AŽ.q be the automorphism group of E. By uniqueness the character is stable under A. This determines an equivalence class of ª ރ s projective representations : A GL 2 nŽ.whose restrictions to U InnŽ.E are not projectively equivalent to ordinary representationsŽ but are lifted by E.. ކ u ¸ s PROPOSITION 0Ž. Griess . There exists a group extension 2 G A AŽ.q lifting . For a proof we refer towx 3, 4 . It follows that the unique monomorphism ކ u ރ ª ކ g 2 * sends G :MŽ.A 2 onto the Griess character g HomŽŽ. M A , ރ* . determined by Žresp. the class of .. In particular g has order 2, as well as its restriction to U. We may identify ރ s ކ s ⌳2 HomŽŽ. M U , * .Hom ŽŽ. M U , 2 . Ž.Uˆ . The image of the restriction 222ކ ª ކ s map Res: H Ž.A, 22H Ž.Ž.U, S Uˆ consists of elements which are fixed by OŽ.q . Using naturality of theŽ. UC sequences from Lemma 0 we therefore get the following. s q COROLLARY. Excluding the case q q2 Ž. E dihedral the image of the 22ކ ª restriction map Res: HŽŽ. A q , 2 .S Ž.Uˆ is generated by the quadratic form q. s q The corollary is false for q q2 . For if E is dihedral of order 8 then AutŽ.E , E, and the quaternion group of order 16 provides for a coun- terexample. In this case the dihedral and semidihedral groups of order 16 496 PETER SCHMID are the unique canonical extension groups of E, and they are representa- tion groups of AutŽ.E . The faithful irreducible characters of the dihedral and semidihedral groups of order 16 have Schur index 1 and the value fields ޑŽ.''2 and ޑ Žy2 . , respectively. s y s q If E is the quaternion group of order 8 Žq q224.Ž., then GL 3 2 S y and the binary octahedral group 2 S4 are the covering groups of AutŽ.E , S4. Here E embeds into the covering groups as the largest normal q 2-subgroup. The faithful irreducible characters of degree 2 of 2 S4 have ޑ y y Schur index 1 and value field Ž.' 2 ; those of 2 S4 have Schur index 2 and value field ޑŽ.'2. In the following we shall always assume that n G 2Ž. without loss . THEOREM 0. The following statements hold: 1 s s q Ž.1HOŽŽ.q , U . 0 unless q q6 , in which case the group has order 2. Ž.2HO2 ŽŽ.q , U .¨anishes for n s 1, 2 and has order 2 otherwise. The first result is due to Pollatsekwx 8 . For n s 2Ž. and for n s 1 the module U is cohomologically trivial for ⍀Ž.q and for O Ž.Žq . Use that q / s y O4 Ž. 2 has a normal Sylow 3-subgroup 1. In the case q q4 the ⍀ , wx module U is the Steinberg module for Ž.q A1 Ž..4 . Griess 3 showed that H2 ŽŽ. O q , U ./ 0 for n G 3, and Sahwx 10 proved that the order of 2 ⍀ s qyG s G H ŽŽ.q , U .is 2 if q q2 n with n 6 and if q q2 n with n 7. With the help of the computer it has been checked that H 2 ŽŽ.⍀ q , U . s q q s y s has order 2 for q q810, q and for q q2n, n 3, 4, 5, 6. Moreover, < 2 < s s q 2 ⍀ s HOŽŽ.q , U . 2 for q q6 Žbut H ŽŽ.q , U . 0 in contrast to the statement in Theorem 1 ofwx 3. . Using the inflation᎐restriction sequence one infers fromŽ. 1 that 1 ⍀ s / q H ŽŽ.q , U . 0 for q q6 . It follows that then the restriction map HO2 ŽŽ.q , U .ª H 2 ŽŽ.⍀ q , U .is injective, which completes the proof of Theorem 0. The author is indebted to Derek Holt for carrying out the computations Ž.on the basis of his GAP cohomology package . It is impressive that he was ⍀" ⍀y able to handle the ‘‘large’’ groups 10Ž.2 and 12Ž.2. 1. AUTOMORPHISMS Keep the notation and assumptions introduced above. OŽ.q acts Ž ab- solutely.Ž. irreducibly on U Lemma 0 and so has no nontrivial normal ⍀ s q s 2-subgroups.ŽŽ. Of course q is simple unless q q42..Ž Hence U O A. is the largest normal 2-subgroup of A s AŽ.Žq as we ruled out the s s dihedral group.Ž.Ž. . The action is faithful so that CUA U and ZA 1. EXTRASPECIAL 2-GROUPS 497 Recall that the group of all automorphisms of A centralizing U and Žhence . O Ž.q , the group of equivalences of U u A ¸ O Ž.q onto itself, is naturally isomorphic with the group Z1ŽŽ.O q , U .of crossed homomor- s 1 phisms. From CUAŽ. U it follows that HŽŽ. O q , U .maps injectively into OutŽ.A . THEOREM 1. The natural map HO1ŽŽ.q , U .ª Out ŽŽ.. A q is an isomor- s s q phism. In particular, OutŽŽ.. A q 1 unless q q6 . Proof. Let ␣ be an automorphism of A s AŽ.q . Then ␣ acts on U and on OŽ.q . For x g O Ž.q and u g U we describe these actions by writing x ␣ and u␣ for the images. The action of OŽ.q on U is also written Ž.u, x ¬ ux. Since this latter action is induced by conjugation in the group A, we have ux ␣ s u␣y1 x␣ . The same relation holds true for the action on the dual module Uˆ and the Ž.diagonal action on Uˆˆm U. Clearly the submodule L , ⌳2 Ž.Uˆ of the tensor square is stable under ␣. Hence we get a similar action on S2 Ž.Uˆˆˆs U m UrL. Thus we have q s qx ␣ s q␣y1 x␣ for all x g OŽ.q . It follows that q␣y1 is left fixed by OŽ.q . By Lemma 0 we must have q␣ s q. Consequently ␣ induces on U an orthogonal transformation, say .   ␣ There is an inner automorphism 0 of A inducing . Replacing by y1␣ ␣ ␣ 0 we may assume that acts trivially on U. But then also s r centralizes OŽ.q A CUA Ž .. This completes the proof. COROLLARY.OutŽ.A acts semiregularly on the set of all elements in 2 ކ H Ž.A, 2 restricting to q. Extensions belonging to the same orbit are repre- sented by isomorphic groups. Proof. We have seen that the automorphisms of A s AŽ.q preserve q. Inner automorphisms act trivially on the cohomology. Let ␣ be a noninner s q ކ u ¸ automorphism of A Žso that q q62.. Let G A be an extension whose cohomology class restricts to q; such an extension exists by the Corollary to Proposition 0. Then we may identify E s EqŽ.with the s inverse image in G of U, and E O2Ž.G . Since ␣ stabilizes q it may be lifted to an automorphism of E. There is no automorphism of G lifting ␣. Otherwise this could be altered by an inner automorphism of G such that it gets the identity on E. But an r s automorphism of G centralizing E must centralize G CEGŽ. A too. On 2-cocycles the action of ␣ may be described by Ž.f, ␣ ¬ f ␣, where fx␣ Ž ␣, y ␣ .Ž.s fx, y for all x, y g A. Write G and G ␣ as crossed products 498 PETER SCHMID ކ ␣ ¬ of A and 2 to the factor sets f and f , respectively. Then Ž.x, c f ␣ ; ␣␣ ␣ ª ␣ Ž.x , c f is an isomorphism G G which induces . Hence G and G are isomorphic groups but represent nonequivalent central extensions of A. The corollary clearly is interesting only in the case OutŽ.A / 1. Never- theless it will be useful for computing the Schur multiplierŽ also in the Appendix. . 2. SCHUR MULTIPLIERS As already mentioned MŽŽ.. O q is trivial up to some few cases, and at any rate it is an elementary 2-groupŽ. Steinberg . Of course M Ž.U s ⌳2 Ž.U ކ is also an 2-space. THEOREM 2. The multiplier MAŽŽ..q is an elementary 2-group. The natural map MAŽ Žq ..ª MO Ž Žq .. is epimorphic and its kernel has order 2 и Inf d2 ª 21ކ ª ª ª 3ކ 0 HOŽ.Ž.q , 2 Ker Ž Res .HOŽ. Ž.q , Uˆ HOŽ.Ž.q , 2 . 22ކ ª s 1, 1 Here Res: H Ž.A, 22S Ž.Uˆ is the restriction map, and d d2is the map obtained by taking cup products with the cohomology class of the group extensionŽ cf.wx 5, Theorem 6.4. . Injectivity of the inflation map on the left means that the dual map MŽ.A ª MO ŽŽ..q is surjective. r Ј ކ ª; r Ј ކ In view ofŽ. UC , Ext ŽŽ.Ž.. O q O q , 22Ext ŽA A , .embeds into KerŽ.Ž Res via inflation . . We may divide out these Ext-groups in the above sequence. ކ u ރ The unique monomorphism 2 * gives rise to an embedding of this modified exact sequence into the analogous one with theŽ. trivial coeffi- ކ ރ cients 2 replaced by *. We get a commutative diagram where the ކ , vertical map on the left is an isomorphismŽ Hom Ž M Ž O Žq .., 2 . HomŽ M Ž O Ž..q , ރ* .. and where the groups H1Ž O Ž.q , Uˆ .and H1 Ž O Ž.q , HomŽ..U, ރ* are identified. From the Corollary to Proposition 0 it follows 2 ކ r that H Ž.Ž.A, 2 Ker Res has order 2, and a corresponding statement holds with respect to ރ*. EXTRASPECIAL 2-GROUPS 499 Suppose we have shown that d2 is the zero map. Then we may infer that ކ , ރ HomŽ. MŽ.A , 2 HomŽ. MŽ.A , * has the order 2 и 3. CANONICAL EXTENSION GROUPS We construct group extensions to the canonical coupling OŽ.q , Out ŽE . and show that different equivalence classes are represented by nonisomor- phic groups. The approach will not depend on Theorem 2. THEOREM 3. There are < 2 ކ < It remains to show that different equivalence classes lead to nonisomor- phic groups. Suppose H and G represent canonical extensions of OŽ.q by E. Identify O ŽH .s E s O Ž.G and HrZE Ž.s A s GrZE Ž.. Assume that there is 22; a group isomorphism : H ª G. Then induces automorphisms on A and on E. By Theorem 1 we may alter by an inner automorphism of G such that it induces the identity on OŽ.q and on U s ErZE Ž .. But now , 1 ކ s induces on E an inner automorphismŽŽ. Inn E Z ŽU, 2 .Uˆ .. We may modify by an inner automorphism of G Ž.induced from E such that it is the identity on both E and OŽ.q . In other words, now describes an equivalence between the extensions E u H ¸ OŽ.q and E u G ¸ O Ž.q . g 2 ކ Proof of Theorem 2Ž. continued . Suppose H Ž.A, 2 is a cohomol- ogy class such that ResŽ. s q. Then the elements in the coset q KerŽ. Res are in 1-1 correspondence with the equivalence classes of Ž central . ކ u ¸ group extensions 2 G A leading to canonical extensions of OŽ.q by E.Ž The classes of the canonical extensions are in 1-1 correspondence with q 2 ކ the elements in InfŽ HŽŽ. O q , 2 .... By the Corollary to Theorem 1 there is a semiregular action of OutŽ.A on the coset q KerŽ. Res where each orbit leads to extensions repre- sented by isomorphic groups. Hence combining Theorems 1 and 3 we obtain that << In view of the Hochschild᎐Serre sequence this completes the proof of Theorem 2. As a consequence the cup product H1ŽŽ. O q , Uˆ .= 23ª ކ HOŽŽ.q , U .HO ŽŽ.q , 2 .is the zero map. 4. REPRESENTATION GROUPS We will have to compute value fields of irreducible characters. The following lemma is only needed for the prime p s 2, but it is perhaps enlightening to state it slightly more generally. Let F be a subfield of the complex numbers. LEMMA. Suppose H is a normal subgroup of some finite group G such that<< GrH s p is a prime. Let g IrrŽ.HbeaG-stable character, with FŽ. s F, and let g IrrŽ.G be an extensionŽ. which exists . Then either FŽ. s F Ž. for some pth root of unity, , or F contains all pth roots of p unity and FŽ. s F Ž' c . for some c g F* _ F*.p EXTRASPECIAL 2-GROUPS 501 This is a very special situation of Clifford theory. One associates to g 2 r , FG the Clifford obstruction FGŽ. H Ž.G H, F* which vanishes if and only if there is Ј g IrrŽ.G extending such that FŽ.Ž Ј s F e.g., seewx 11, Lemma 9.Ž.Ž. . Then F s F as claimed. Otherwise the order of wx FGŽ.is p; hence F contains the pth roots of unity by 11, Theorem B . Now use that the cup product with a generator of H 2 Ž.GrH, ޚ gives an isomorphism from Hˆ 0Ž.GrH, F* s F*rF*toHp 2 Ž.GrH, F* , and use functoriality of FGŽ.with regard to F. Note that the Ž character . field p FcŽ.' obtained is independent of the generator chosen and is the same for all characters of G extending . Recall that an extension group, G, to the canonical coupling of OŽ.q with E s EqŽ.is a representation group of A s A Ž.q if there is a character g IrrŽ.G extending . The field ޑ Ž. of character values is independent of the chosen extension since GrGЈ is elementary of order 2 s q or 4, the latter only occurring for q q4 . THEOREM 4. There are just Proof. Consider the Griess character g g HomŽŽ. M A , ރ* . . By Propo- sition 0 it has order 2Ž which of course now is also a consequence of ކ u ¸ Theorem 2. . A central extension 2 G A gives rise to a representa- tion group if and only if G maps onto g . Then the cohomology class of G restricts to q, and G is a canonical extension group lifting some projective ރ-representation of A associated to . It follows fromŽ. UC that there are as many different equivalence classes of such central extensions as there are elements in the group r Ј ކ , r Ј ކ ExtŽ.Ž.Ž.A A , 22ExtŽ. O q O q , , s q which has order 2 unless q q4 . Application of Theorem 3 yields that distinct classes are represented by nonisomorphic groupsŽ since the differ- 2 ކ ence of two classes is inflated from HŽŽ. O q , 2 ... Now let G be a representation group, and let g IrrŽ.G be a character s extending . Then is faithfulŽŽ.Ž.. as CEG ZE . It is well known how to compute the values of Ž.`a la Hall᎐Higman . Every coset of E in G / < < 2 s < < contains an element x with Ž.x 0, and then Ž.x CxE Ž.: ZE Ž . s < < wx CxU Ž.Žsee 9, Lemma 1.1 for a lucid proof. . More precisely, we have Ž.x s и 2 sŽ x. 502 PETER SCHMID g 1 ޚ g ޚ for some 4th root of unity, , and some sxŽ. 2 , where sxŽ. if and only if x belongs to the inverse image in G of ⍀Ž.Žq cf.wx 4, p. 144. . Let s Ј s : ⍀ H G and let H . Then E H, and H is the inverse image of Ž.q s q in G unless q q4 . Since is rational valued and, therefore, all Galois conjugates extend and are of the form s и for some linear character of G, we see that ޑŽ. s ޑ. s s y Suppose F is a splitting field for with F FŽ..If q q2 n is of negative type, this requires that y1 is a sum of two squares in F.Ž That m ޑŽ. s 2 here may be seen from the Frobenius᎐Schur count. . By an elementary property of Schur indices m F Ž.is a divisor of и и s m FEŽ.Ž, .F Ž, .: F Ž. 1. Hence there is aŽ. faithful, absolutely irreducible FG-module V affording s ( . Embedding G into GLŽ.V we see that the normalizer NEGLŽV . Ž. G C is a central product over ZEŽ., C , F* being the group of scalar multipli- cations on V.Ify1 is not a square in F, then G is the unique central factor of C Ž.Krull᎐Schmidt . Otherwise there are< HomŽŽ..GrGЈ, ZrZE < s s q 2Ž resp. 4 for q q4 . such factors. Evidently these are representation groups of A s AŽ.q . Since they differ from G by commutator extensions r Ј ކ classified through ExtŽ.A A , 2 , they represent the pairwise nonisomor- phicŽ. though isoclinic representation groups. / q g _ / Suppose first that q q4 . Choose x G H with Ž.x 0. Then Ž.x s ''2or y2 , and ޑŽ .s ޑ Ž Žx .. by the lemmaŽ since H is the inverse image in G of ⍀Ž.q and is rational valued . . Moreover, then s j 2 sy G0 H Hxi is the other representation groupŽŽ. within GL V ; i 1.. ޑ s ޑ If 00is the character of G afforded by V then Ž. 0 ŽŽ..i x . s q r It remains to examine the case q q4 . Here G H is a fours group. Let G01, G , and G 2be the three distinct subgroups of G with index 2, say G0 ⍀ g _ / being the inverse image in G of Ž.q . Choose x jjG H with Ž.x j0. s " " s " " y s Then Ž.x0 1or i and Ž.x j ''2or 2 for j 1, 2. If ޑ s ޑ s ޑ Ž.x01is rational, then Ž . Ž Žx .. Ž Žx2 .. by the lemma. Otherwise we have ޑ s ޑ s ޑ s ޑ i r4 Ž.Ž.Ž.i, Ž.x12i, Žx . Že .. Passing to isoclinic variants of G ŽŽ..within GL V we see that both s situations happen, the first one with both options on Ž.x j , j 1, 2. The 8th cyclotomic field ޑŽ.e i r4 occurs twice. s s y COROLLARY Ž.of the proof . The Schur index m ޑ Ž.1 unless q q2 n is of negati¨e type and ޑŽ. s ޑ Ž'2, .in which case the index is 2. EXTRASPECIAL 2-GROUPS 503 5. MINIMUM DEGREE REPRESENTATIONS ␦ Suppose X is a finite group. Then let 0Ž.X denote the minimum of the degrees of all faithful characters of X.Ž A faithful character of X of ␦ degree 0Ž.X need not be irreducible, in general. . Similarly denote by ␦ ރ y cŽ.X the minimum degree of the faithful projective Ž.representa- wx ␦ tions of X. Landazuri and Seitz 7 examined cŽ.X for groups X of Lie ␦ F ␦ wx type. Clearly cŽ.X 0 Ž.X and, following Feit and Tits 2 one may ask ␦ whether cŽ.X may be decreased if one allows noncentral extensions of X. So define ␦Ž.X as the minimum of the degrees of the faithful representa- tions of all the finite groups having X as epimorphic image. wx ␦ - ␦ It is shown in 2 that if X is a simple group and Ž.X c Ž.X then ␦Ž.X is a 2-power. This uses the classification of the finite simple groups Žwhich at the time of publication ofwx 2 has been almost completed. . G s q ␦ ⍀ s n s LEMMA. Assume that n 4 and exclude q q8 . Then ŽŽ..q 2 ␦ŽŽ..O q . It is obvious that ␦Ž⍀ Žq ..F ␦ ŽO Žq ... By Theorem 4 we have ␦ ŽO Žq .. F 2n . By Landazuri and Seitzwx 7 every faithfulŽ. projective representation of ⍀Ž.q has degree greater than 2n . Note that MŽŽ..⍀ q s 0 by assumption Ž.Steinberg . From the Theorem inwx 2 it follows that ␦ŽŽ..⍀ q s 2m for some integer m F n. Moreover, there is an irreducible representation ⍀ ª s Ž.q Sp2 m Ž. 2 . This forces that m n. G s q u THEOREM 5. Assume again that n 4 and exclude q q8 . Let R H ¸ OŽ.q be a finite extension such that H admits a faithful character, , of degree ␦ŽŽ..O q s 2.n Suppose G is a minimal supplement to R in H. Then GrGЈ is a cyclic 2-group, and GЈ is isomorphic to the unique co¨ering group of AŽ.q Ј. In particular, is irreducible on E s R l GЈ , E Ž. q and Z Ž H . s / CEH Ž. 1. Proof. Let A s AŽ.q , as usual. By virtue of the inflation᎐restriction sequence MŽ.A , M ŽAЈ ., which has order 2 Ž Theorem 2 . . So there is a unique perfect group G* which maps onto ⍀Ž.q and which has ZG Ž* ./ 1, and this G* is isomorphic to the commutator subgroup of any representa- tion group for A. We assert that GЈ , G*. Observe that GЈNrN , ⍀Ž.q is simple. By construction GrGЈ is a cyclic 2-group and N s R l G is in the Frattini subgroup of G.In particular N is nilpotent. It is also evident that G is irreducible. By Landazuri and Seitzwx 7 N cannot be central in G. Let Y be a G-invariant subgroup of N, and let T be the inertia group in g G of some irreducible constituent of Y . By Clifford there is IrrŽ.T s G such that G is induced. We have 2 nTNs Ž.1 s Ž.1 и < If TN s G, then T s G as N is in the Frattini subgroup of G.If TN s GЈN, then ␦ŽŽ..⍀ q F TNŽ.1 s Ž.1 r2, in contrast to the lemma. Otherwise the permutation representation on theŽ. right cosets of TN in G gives rise to a faithful character of OŽ.q of degree < From the knowledge of the maximal subgroups of Sp2 nŽ. 2 one infers that OŽ.q is mapped onto a maximal parabolic subgroup of Sp2 nŽ. 2 of type " wx O2 nŽ. 2 stabilizing a hyperplaneŽ see the Main Theorem in 6, Table 3.5.C. . But this implies that G normalizes some extraspecial subgroup of E of index 2, against our choice of E. It follows that GrC , A. By the exact homology sequence MŽ.A maps onto GЈ l C. From< MŽ.A < s 2 and ZEŽ.s EЈ : GЈ we conclude that GЈ l C s ZEŽ.. We even have E : GЈ since GЈ acts irreducibly on U s ErZEŽ.. Consequently GЈ l N s E and GЈ , G*, as desired. APPENDIX 1q2 n Let E s EqŽ.s 2" be as before, and let E˜s E( Z be the central s , product with a cyclic group Z Z4 of order 4. Identify U with InnŽ.E˜ E˜rZ and the symplectic form q˜ with the commutator form on U defined by E˜˜. Let A s AŽ.q˜ be the group of all automorphisms of E ˜centralizing Z. We have a group extension U u A˜¸ SpŽ.U s Sp Ž.q˜ EXTRASPECIAL 2-GROUPS 505 which is nonsplit for n G 3Ž Griesswx 3.Ž. and n s 2 computer . By Pollatsek wx8HSp< 1ŽŽ..U , U < s 2 unless n s 1, and by Steinbergwx 12 MŽŽ.. Sp U s 0 for all n G 4. ކ u ކ ª The monomorphism 22Z induces the zero map ExtŽ.U, ExtŽ.U, Z . This yields a natural splitting of theŽ. UC sequence for Z.In particular, H 2 Ž.U, Z , Ext Ž.U, Z [ HomŽ. M Ž.U , Z , Uˆ[ ⌳2 Ž.Uˆ as an SpŽ.U -module. Here q˜g ⌳2 Ž.Uˆ is the unique nontrivial element which is fixed by SpŽ.U , and it is the cohomology class assigned to s qy( , ( E˜ EqŽ.Ž˜ cf. Lemma 0 . . We have Eq Ž.2 n Z Eq Ž.2 n Z, and there are twoŽ. complex conjugate faithful irreducible characters of E˜, each being stable under A˜. Let F s ޑŽ.i and let V be an FE-module affording . Embed E into s s , GLŽ.V and let Hq NEGLŽV . Ž.and C CEGLŽV . Ž.. Then C F* consists r , ⍀ of scalar multiplications, and Hq EC Ž.q by Theorem 4. We identify s s ( s r Z ²:Ži and E˜˜E Z.. Let H NEGLŽV .Ž.. Then H C is isomorphic = to a subgroup of A˜, and H Hq for both types of q Ž.positive or negative . It follows that HrC , A˜ except possibly when n s 1. Clearly HЈ l C = EЈ s ZEŽ ..If n G 2 then O Ž.q is a proper subgroup of Sp Ž.q˜ Ј and, by the exact homology sequence, the finite group MŽ.A˜ must map onto HЈ l C s Z. For n G 3 the symplectic group SpŽ.q˜ is perfect Ž simple . . In this case we have a group extension E˜u G˜ s HЈ ¸ SpŽ.q to the canonical coupling ; ˜ SpŽ.q˜ ª A˜˜rInn ŽE .which, in addition, admits a faithful irreducible char- acter of degree 2n having F s ޑŽ.i as value fieldŽ with Schur index 1. . Passing from n to n y 1 by considering the centralizer of a central factor of E of order 8 we see that the same holds true for n s 2, 1. We infer that the image of the restriction map Res: H 2 Ž.A˜, Z ª H 2 Ž.U, Z is generated by q˜ Žcf. the Corollary to Proposition 0. . Showing that AutŽ.A˜˜preserves q˜, Theorem 1 carries overŽ so that Out Ž.A , HSp1ŽŽ..U , U is not trivial for n G 2. . The proofs for Theorems 2 and 3 apply likewise. In particular, the cup product with a generator of the cohomology class of U u A˜¸ SpŽ.U gives rise to the zero map 1, 1 1 ª 3 ᎐ d2 :HŽŽ.. SpU , Uˆ HSp ŽŽ..U , Z in the corresponding Hochschild Serre sequence. We obtain that the multiplier MŽ.A˜ has order 2 for n s 1, is of type = s Z42Z for n 2, 3, and is cyclic of order 4 otherwise. As observed above there exists a representation group G˜ whoseŽ. complex conjugate faithful irreducible characters of degree 2n have value field ޑŽ.i . For n s 1, 2 there is one further representation group yielding the character field ޑ i r4 s " ( s Že .Ž. For n 1 this is the central product 2 S4 Z.. For n 2 there 506 PETER SCHMID are even two more canonical extension groups without admitting charac- ters of this kind. For n s 3 there are two equivalence classes of extensions to the canonical coupling, where one is a representation groupŽ namely that described above. . 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