A cohomological approach to theory of groups of prime power order

PHAM ANH MINH KODAI MATH. J. 17 (1994), 571—584 A COHOMOLOGICAL APPROACH TO THEORY OF GROUPS OF PRIME POWER ORDER BY PHAM ANH MINH §0. Introduction Let G be a finite group and A a G-module. Consider the set of group extensions (Γ) 0-*Λ-»Γ->G-»1 in which the G-module of A defined via conjugation of G coincides with the one already given in A. Two extensions (Γ) and (Γ') are said to be equivalent if there exists a homomorphism / : Γ —»• Γ7 such that the diagram 0 •/ > A ——> Γ7 •/ ——> G is commutative. Let £(G,A) be the set of equivalence classes of such extensions. It is well-known that there exists a natural 1-1 correspondence with 0[Γ] the factor set of the extension (Γ). Good description of H2(G,A) is then an effective tool to the study of group extensions of A by G. This material has been used by several authors: Babakhanian [1], Baer [2], Beyl [4], Evens [7], Gruenberg [9], Schreier [20] [21], Stammbach [22] ... to obtain group theoretical results. In this work, we restrict ourselves to the case where G is a group of prime power order (i.e. a p-group); in such a case, A can be chosen to be central and elementary. Our method is focussed on the Hochschild-Serre spectral sequence of a central extension: by studying the relation between the Hochschild-Serre filtration of H2(G,A) and the Frattini class of Γ, we obtain cohomological proofs of results concerning the Frattini subgroup of a p-group. Most of these results were already proved by other group theorists (Berger-Kovacs-Newman [3], Blackburn [5], Kahn [13] [14], Hobby [10], Thompson [23] This note is organized as follows. In §1, we consider the central extension by an elementary abelian p-group and the term E\£, (i+j = 2) of the Hochschild-Serre spectral sequence for it. §2 is devoted to the study of the relation between the Hochschild-Serre filtration of H2(G,A) and the Frattini class of Γ; the main results of this section are Theorems 2.1 and 2.3. They are applied to the study of p-groups with cyclic Frattini Received May 14, 1993. 571 572 PHAM ANH MINH subgroup and d-maximal p-groups in §3; some of these results were appeared in our earlier papers, we can refer to [6], [16]. From now on, for every p-group G, H*(G),Z(G), Φ(G) denote respectively the modp cohomology algebra of G with coefficients in the prime field Zp, the center and Frattini subgroup of G. §1. Central extension by an elementary abelian p-group: Let G be a p-group and A a central and elementary abelian subgroup of G of rank n. Consider the central extension with K = G/A. We denote by z the factor set of the extension (G). So z is of form z = (zί,...,zn)€H*(K,A) = $ H\K) n times with Zi G H2(K). Let { ΐ / i , . . . , un} be a base of Hl(A) and V{ = βui with β the Bockstein homomorphism. Then ^ { E[uι,...,un]®Zp[vι,...,vn], ifp>2 with ϋ?[x,2/,...] (resp. Zp[x,y,...]) the exterior (resp. polynomial) algebra with generators x , t / , . . . over Zp. We denote by {Er(G),dr} the Hochschild-Serre spectral sequence (HSss, in short) for the extension (G). So E2(G) = JT(#) ® H*(A) => H*(G). The base {ι/ι,..., un} of H1(A) can be chosen such that d%(ui) — z». Note that κ, , v^ are transgressive and ds(vi) = /?2, . Generalizing [17, Prop. 1.5] and [18, Prop. 2.1], we have . PROPOSITION 1.1. (a) E*?(G) = E^(G) = JΪ 2 (ff)/(*ι, -,**); (c) If zi,..., zn are linearly independent in H2(K), then the vector space over Zp generated by v ι , . . . , v n , and E^2(G) = E%)2(G). Furthermore, Vi G £&2(G) iffβzi = 0 mod(zι,. ..,zn) in H*(K). Proof. We need to prove that E^>2(G) = ZP{VI, . . . , v n } if z i , . . . , zn are linearly independent in H2(K). The remaining parts of the proposition are obvious by noting that dr is of bidegree (r, 1 — r). Since v, is transgressive, we have v, G E3* (G). Let u = 5^λtι, ιy be an element of Kerc?2j with λ^ G Zp. By a change of base of A, we can assume that u = 53^2)b-i2A:^2A;-ιW2A: So 0 = ^λ2fc-i2fc(^2A;~ι^2Jb ~ ti2fc-ι*2fc) Since z i , . . . , 2:n are linearly independent, we have \2k-i2k — 0. The proposition follows. \\ A COHOMOLOGICAL APPROACH TO THEORY OF GROUPS 573 Let {F*H*(G}} be the Hochschild-Serre filtration on H*(G). Following [11], we have We have then an isomorphism between vector spaces over Zp: (1.2) H\G)] -?-+ J&°(G) Θ E£(G) Θ 1 The map η is defined as follows: if x G F1H2(G) (o < i < 2), then ηx =pr a?, with pr the projection map from FiH2(G) onto E^"l(G). Note that the extensions (G) is obtained from successive central extensions by Zp. In fact, let {αι,...,α n } be the base of A, dual to {tiι, . . ,tι n } Set G^ = G/ < αj,αj+ι, . . . ,α n >. So G^1) = /if and G^n^ = G. We get then a sequence of central extensions 0 -+< α, >-> G('+1) -> G« _+ i with factor set z, G Im (1.3) We now consider the simplest case where n = 1 and z ^ 0. Set yi =< a >, w the dual of α and v = βu. Let Jf be an arbitrary element of H2(G) and Γ the central extension (Γ) 0 -> Zp —^-> r -» G -* 1 with factor set X . Set Z — iZp . Note that Γ is isomorphic to the group whose underlying set is Zp x G, and the multiplication is given by with a,beZp,g,heG.We have PROPOSITION 1.4. (a) Z C Φ(Γ) iff X φ 0 in #2(G). Furthermore, if X ± 0, (b) J/iyX = v, /Aaw (0,αX = (1,0); (c) If ηX = x ® ι/; it zίΛ x G H1(G)} and g an element of G with x(g) = 1, Proo/. (a) It is obvious that X = 0 implies that Γ = GxZ, so Z tf_ Φ(Γ). Conversely, assume that Z (£ Φ(Γ), then there exists a maximal subgroup H of Γ such that Z (£ H. Hence Γ = H.Z = # x Z, so the extension (Γ) splits. This implies X = 0. For X ί 0, since Γ/Φ(Γ) is elementary abelian and Γ/Φ(Γ) Si Γ/Z/Φ(T)/Z = G/Φ(Γ)/Z, we have Φ(G) C Φ(T)/Z. On the other hand, let I be normal in Γ with L/Z = Φ(G), then Γ/L S T/Z/L/Z = G/Φ(G). Since G/Φ(G) is elementary abelian, we have Φ(Γ) C I. Hence Φ(Γ) = Z, so Φ(Γ)/Z = Φ(G). (b) Let B be the subgroup of Γ with B/Z = A> we have the central extension Q -> Z -^ B -+ A->1 with factor set Res(A, G)X = v G H2(A). So B - Cp2, the cyclic 2 group of order p . Hence (0,α)* = (1,0). (c) Let G be the subgroup of Γ with C/Z =< α, 0 >, we have the central extension 0 -» Zp -> G ->< 0,51 >-* 1 with factor set Res(< a,g >,G)X = x.u G H2(< a,g >). So[(0,0),(0,α)] = (l,0). D We complete this section by some remark concerning the map η defined in (1.2). 574 PHAM ANH MINH Remark 1.5. (a) The map η is closely related to the Genea one in [8], and especially to the comodule structure of H*(G) over H*(A), defined by Stammbach in [22], as follows : since A is central, the map m : A x G —» G, (α,y) H-> a.g is a homomorphism. One gets then a map µ : H*(G) —> H*(G) 0 H*(A) (the Genea map is just the restriction of µ to Hl(G) -» Hl(G) ® Jϊ1^))- Hence> we have Λe map 2 2 µ2 : H (G) -> H (A) 0 (H\G) ® J One can check that µ 2 is nothing but (b) As Im lnί(H2(K) -+ H2(G)) = £&°, the sequence 0_ i(K) H is exact, with η1 the restriction of 77 to E^(G)®E^(G) and H2(G)A = KerRes(#2(G) —>• jy 2 (Λ)). We have then the extending Hochschild-Serre exact sequence, which has been given in [15] and [24]. §2. Frattini filtration of a p-group and HSss Let G be a p-group. Set Φ°G - G,Φ X G = GP.[G,G] (the Frattini subgroup of G), . . . ,Φ*+1G = ((Φ2G)P.[Φ*G,G],____We have then the following descending sequence of subgroups of G G = Φ°G D Φ X G D D Φ'G D Φ l+1 G D . . . . It is called the Frattini filtration of G. The Frattini class of G, denoted by c/φ(G), is defined to be the smallest integer m such that Φm(G) = {!}. For example, c/φ(G) = 0 iff G = {!}, c/φ(G) = 1 iff G is elementary abelian, c/φ(G) = 2 iff G is almost-special ( i.e. a central extension of an elementary abelian p-group by an another). It is clear that G/Φ*G is of Frattini class i. By setting AiG = Φί""1G/ΦlG, AiG is then a vector space over Zp. In the remaining of this section, we suppose that c/φG = m + 1. So A m +ιG = Φ m G is central and elementary abelian. Furthermore, there exists quotients Gi of G given by the central extensions (G2) 1 -> ^2G -* G2 -* ^iG -* 0 (G3) 1 -* A3G -> G3 -> G2 -^ 1 (Gm+ι) 1 —>• A m +ιG —>• Gm+ι —»• G with Gm+ι = G, Gi-ι = Gi/ AiG and Gi = AiG. Let n, -dim ZpAiG and ,« - (,«, . . . , z«+1) G +i « i+l time. H2(Gi) be the factor set of central extension (G, +ι). G is then determined by the sets {AiG}ι<ι<m+ι and {z^}ι<2<m The groups Gz , together with the maps Inf(Gj,G<) (j > 0 f°rm a direct system. For simplicity, we denote by Inf2(G2 ) the image of the map A COHOMOLOGICAL APPROACH TO THEORY OF GROUPS 575 We now consider a central extension (Γ) 0 -> Zp -> Γ -+ G -> 1 2 with factor set X G H (G). We want to know the Frattini class of Γ. It is clear that m - ί - 2 > c / φ Γ > r a - f l . The following is obvious from Proposition 1.4 and its proof. THEOREM 2.1. (a) If X g Inf2(Gm), then c/φΓ = m + 2; (b) //JT G Inf2(Gi), *Λen c/ΦΓ = m + 1 and —A^+iΓ ~ nt +ι + 1 z/X, -4 , . . . , ^nt+ι = ni + 1 if X, £j , . . . , %$+! are αre near H ly independent, linearly dependent. We have then COROLLARY 2.2. z^,..., z$+ι are linearly independent in #2(G;)/Inf2(Gj, G;_ι). Let G and Γ be given as in (1.3). Assume furthermore that z φ 0 and X / 0. We have THEOREM 2.3. (a) IfηX G El'l'(G), then E^2(T) = 0; Proof, (a) If τ?X G E^l(G), then ryX is of form x ® ti, with x G ff1^)- Let ^ be an element of K such that x(g) φ 0 and (D) the central extension 0 —>• Zp —*• D —»•< ^ >~^ 1. Then Res(£>, G)X H^ x 0 u G El^(D}. Hence /?X H+ x ® /?w G E^2(D) which is non-zero. Since x <g> /?tt is not of form y.x 0 /?u with y G Hl(< g >), it follows that βX = d3(βu) / 0(mod X). This implies f^2(Γ) = 0. (b) Obvious by noting that v is algebraically independent in Eoo(G). \\ From this, we obtain the following, which has been proved by Kahn in [14] COROLLARY 2.4. Let i > 1 be an integer. The following assertions are equivalent: (a) (Φ 'G^ = Φl+1G; (b) (Φ*G)^ = Φ*+1G; V k > i; (c) ^1(G,-) = 0; (d) E#(Gk) = 0, V Jb > , with (Gj) ^Λe extension 1 —> ^42G —» G2 —* Gi_i —>• 1. §3. Some applications We are now going to apply our results to obtain cohomoligical proofs of some group theoretical one. We are interested in p-groups with cyclic Frattini subgroup and dmaximal p-groups. a. p-groups with cyclic Frattini subgroup. We give here cohomological proofs of Hobby's theorem [10] (III 7.8 c in [12]), which asserts that Φ(G) is cyclic if Z(Φ(G)) is 576 PHAM ANH MINH cyclic, and of Berger, Kovacs and Newman's result [3] on the classification of p-groups with cyclic Frattini subgroup. First, we prove THEOREM 3.1 (Hobby [10]). If Z(Φ(G)) is cyclic, then so is Φ(G). We need LEMMA 3.2. Let G and Γ be given as in (1.3). Assume that X φ 0, then: (a) Every extension of a subgroup of Φ(G) Γ\Z(G) by Z is contained in Z(Φ(T)); (b) IfηX G #3'0(G), then Ax Z is a subgroup ofΦ(T) Π Z(T). Proof, (a) Let α G Φ(G) Π Z(G) and 6 G Γ such that bG = a. For g,h G Γ, since [#, 6] and [Λ, 6] belong to Z, we have [0P, 6] = 1 and [[#, Λ], 6] = 1 in Z. (a) is then proved. (b) Obvious from the definition of the Hochschild-Serre filtration on Bar cochains. o From this and Proposition 1.4, we obtain LEMMA 3.3. With the assumption of Lemma 3.2, assume that Z(Φ(T)) is cyclic, then Φ(G) Π Z(G) is cyclic and ηX G JE LEMMA 3.4. With the assumption of Lemma 3.2, assume thatΦ(T)ΠZ(T) and E%?(Γ) ± 0, then Φ(G) Π Z(G) is cyclic and ηX G E is cyclic Proof. Consider the extension (G), with A an arbitrary subgroup of Φ(G) Π Z(G) of order p. Since Φ(Γ) Π Z(Γ) and E%?(T) ^ 0, it follows from Lemma 3.3 that ηX G E%?(G). So Res(^} G)X — v. Since A is an arbitrary normal subgroup of G of order p, Φ(G) Π Z(G) is cyclic. The lemma follows. Q Proof of Theorem 3.5. Let |G| = pn+/ and |Φ(G)| = pl. By Lemmas 3.3 and 3.4, we get a sequence of central extensions (Gi)0 —>• Zp —> G» —>• Gt +ι —>• 1, 1 < iI < /, with GI = G,G/+ι = C^, and the factor set z% of (G<) satisfies ^ G £^2(G<+ι), /?^ = 0. Hence Φ(G/), Φ(G/_ι), ..., Φ(Gι) are cyclic. The theorem is proved. The classification of p-groups with cyclic Frattini subgroup has been done by Berger, Kovacs and Newmann [3]. These groups can be constructed cohomologically as follows. First, we recall some basic facts of cohomology of groups. Let 6 be a generator of the cyclic group Cpι and ub , vb be respectively the 1- and 2-cocycles of Cpι given by l tii + j <P . otherwise. So vb = βub, for / = 1. It is well-known that (3.6) H*(Cpl) = lZM p E[uι]®Z[vι>\ϊ]} if/ = otherwise. A COHOMOLOGICAL APPROACH TO THEORY OF GROUPS 577 Hence, if k is an integer and Cpι x C*"1 =< 61, ...ΛΛi' = ^ = [fyΛ] = 1,2 £ i ^ k,l < j ^ k >yby setting tιt = t/ ό ., v< = v&., we get H*(Cpι xC*- 1 )- ^ £[tiι]®Z 2 [vι]®Z 2 [ti2,...,t4]b], ,u*;l]®£p[vι, ,v*;2], if/> 1 andp=2 if p> 2 The following is then obvious. LEMMA 3.7. Let 0 / X G H2(Cpι x C*"1). Assume furthermore of l υ 1 1 >, Cpι x Cp~ )X = bp - that Res(< t / / > l . TΛen X can δe reduced by an automorphism of 1 Cpι x Cί" to one o/rte canonical forms /=ΐ //— ^ or ^; anrf 1 0 or 1, otherwise. Let £,_ I > ^,-/ , , , , rfp \DS, = 2. /+2 and M be an extra-special p-group of order p or one of the groups given by (1) M(p'+2) =< α,6/α^41 = 6" = l,α* = α1+ί>l > for p > 2, (2) M(2<+2), D(2/+2) =:< α,6/α 2 = ό2 rr 1,6-^6 = α"1 >, / 2 2l+1 Q(2 + )rr<α,ό/α = 6 2 ,6~ 1 α6 = a"1 >, 2 2 +1 5(2<+ ) =< α,6/α ' = 62 = 1, 6-^6 = α"^2""2 > (especially, D(8) - M(8), 5(8) = C4 x C2). We have LEMMA 3.8. Let 0 —> Zp —>• G —>• Cpι x C*"1 —> 1 δe α central extension with factor set 0 ^ z G H2(Cpι x CjJ""1) having one of the forms given in Lemma 3.7. TAen G z$ isomorphic to one of the following groups m—1 times m —1 times αnrf in t/ Aαf follows, A 5 means the central product of A and B with \AΓ\B\ = p. Proof. Obvious from the fact that the factor sets of the central extensions 0 —» Zp —» Cpi+i —» Cpj —> 1, 0 -> Zp -»• G -f Cpi x Cp -> 1 W w with G = EjQs (for / = 1), Mpz+2 are respectively vj, 1/1.1/2, w f + 1/2 + ι 2 5 vι + t/ι.ti2 D 578 PHAM ANH MINH Analogous results can be stated if we replace Cp\ by D(2l). Recall that D(2l) =< α, 6/α2'"1 = b2 = 1, α 6 = α"1 > . Let ua,ub be elements of Hl(D(21)) given by ua(alV) - i,ub(alV) = j for 0 ^ i < 2 / ~ 1 ,0 ^ j < 2 and Z/ G H2(D(21)) the factor set of the central extension 0 -> Z2 —> -> D(2') -> 1. The following is due to Quillen [19] and Mui [18]. LEMMA 3.9. H*(D(21)) = P[ua,u^Z{\/(u2a + tιβ.tn) . Furthermore, we have: (i) 0Z/ = tι6.Z,._a (ii) if A =< a2 yalb > is a maximal elementary abelian subgroup of D(2l), with 0<i< 2'-1, then Έ^s(A,D(2l))Zt = 2 ί _ 2 + ua2ι-2.uaib. Let c be a generator of 62- Set Γ = D(2l) x 62, we have LEMMA 3.10. Let X G #2(Γ) w^Λ Res(< α 2 '" 2 >,Γ)X = u 2 2 ,- 2 . ΓΛen X can be reduced by an automorphism ofT to one of the canonical forms with µ G Z2. Proof. [16]). D The proof follows by appropriate change of generators of Γ (for details, see Let k be an integer and {&ι,... A-i} a base of C*"1. Set Φ f = D(2l) x C^"1, then if*(ψj) ir P[u α ,lί δ , W 1 } ...,!/£_!, Z/]/(lί2 + ^α Wδ). By appropriate change of generators of Φ/, we get LEMMA 3.11. Let X G # 2 (Φ/) ti ΛA Res(< α 2 '~ 2 >,Φ/)X = « 2 2ί - 2 . TAew ^ can be reduced by an automorphism o/Φ/ to one of the canonical forms m-l ι=l m-l z/ + tι; + tιa.t4ι 2 i=2 m-l Z] + U2a + Ua U\ -\-U\-\- 1, 2/ G ^2 - Let if = D+(2ί+2) (resp. Q+(2'+2) be the central extension with factor set Z\ + u2a + w α ^ι (resp. Zι + u2a + ua.uι + u2). Since the factor sets of the extensions A COHOMOLOGICAL APPROACH TO THEORY OF GROUPS 579 with G = £>(2/+1), 5(2/+1), Q(2/+1) are respectively Zh Z} + u 2 , Z} + tij, (see e.g. Mui[18]), we have LEMMA 3.12. Let 0 —> Z^ —» G —* Φ/+ι —>• 1 be a central extension with factor set 0 φ z € #2(Φ/+ι) having one of the forms given in Lemma 3.11, then G is isomorphic to one of the following groups m — 2tιmes 2 2 u Λere AT 25 either D+(2'+ ), Q+(2'+ ), £>(2<+2) - G4 or S(2'+2) - G4. Berger-Kovacs-Newman's result can be stated as follows. THEOREM 3.13 (Berger, Kovacs, Newman [3]). Let G be a p-group with cyclic Frattmi subgroup of order pl . If\G\ = pn+ϊ , then G is isomorphic to one of the groups given in Lemmas 3.8 and 3.12. To prove it, we need LEMMA 3.14. Let (G) be the central extension given in (1.3). If K is not elementary abelian, then Φ(G) is cyclic iff Φ(K) is cyclic and z \-+ av G E^2(G) with 0 ^ α G Zp. Proof. By Proposition 1.4 a), we have the central extension Q —> A —+ Φ(G) —» Φ(K) -» 1. So Φ(G) is cyclic iff Φ(K) is cyclic and Rεs(Φ(K),K)z ^ 0. The lemma follows. \\ LEMMA 3.15. LetK = CpιxCp~l orD(2 / )xC|~ 1 and G be one of the groups given m Lemmas 3.8 and 3.12. ΓΛen £"^2(G) ^ 0 iff G ^ Cpι+ι x C*-1 or £>(2/+1) x C^'1 Proo/ It is. obvious that f&2(G) ^ 0 iff ^ = 0(mod z) in H*(K). This fact is equivalent to z = v\ or z = Z\. The lemma follows, p Proof of Theorem 3.16. We proceed by induction on /. The theorem is clearly true for / = 1. Assume that it holds for / — 1 (/ > 2). Let Z be the subgroup of Φ(G) of order p. By Lemmas 3.14 and 3.15, Φ(G/Z) is cyclic, G/Z = Cpι x C%~1 or D($) x G^1 and the factor set for the central extension 1 —* Z —> G —» G/Z —»• 1 is of one of the forms given in Lemmas 3.7 and 3.11. The theorem follows from Lemmas 3.8 and 3.12. b. rf-maximal p-groups. For every p-group G, let d(G) = dim^Γ1(G) the minimal number of generators of G. Following Kahn [13],, G is said to be d-maximal if d(K) < d(G) for every proper subgroup K of G. These groups have been studied by Blackburn [5], Kahn [13][14] and Thompson[23]. The following is due to Blackburn (for d(G} < 3) and Thompson (for p > 2 and d(G) replaced by |G/[G,G]|) and reproved by Kahn in [13]. 580 PHAM ANH MINH THEOREM 3.17 (Blackburn-Thompson). Let G be a d-maximal p-group, then G is of (nilpotence) class < 2 provided that p > 2, or p = 2 and d(G) < 3. It is reasonable to ask whether the conclusion remains true for p = 2 and d(G) > 4. In [13], Kahn claimed that it is valid for d(G) = 4, but this claim is not true. We prove THEOREM 3.18. Let G be a d-maximal p-group, then G is of class 2 and Inf (H2(G/Φ(G)) —> H2(G)) is surjective, provided that one of the following conditions is satisfied: (a)p>2. (b) p - 2 and |Φ(G)| = 2m with m < 2. COROLLARY 3.19. // G is d-maxvmal, G is of class 2 provided that one of the following conditions is satisfied: (a) p > 2 . (b) p=2 and |Φ(G)| < 23. (c) p=2 andd(G)<3. (d) Φ(G) is elementary abelian. THEOREM 3.20. For every n > 4, there exists a d-maximal 2-group Q\ of class > 3 with d(0\) - \. From those, we obtain another proof of Theorem 3.17, and the fact that any dmaximal p-group G is of class 2 iff Φ(G) is elementary abelian. Besides, Theorem 3.20 shows that the mentioned conclusion is false for p = 2 and d(G) > 4. We need the following lemmas. The first one is due to Kahn [13]. LEMMA 3.21. If G is d-maximal and N is normal in G, N C Φ(G), then G/N is d-maximal A cί-maximal p-group of class < 2 is almost special, according to the following. LEMMA 3.22. If G is d-maximal, then Φ(G) = [G,G\(the commutator subgroup of G). Hence, if G is of class 2, Φ(G) is elementary abelian. Proof. By Lemma 3.2, G/[G, G] is d-maximal. Since G/[G, Gl\ is abelian, it is elementary abelian. So Φ(G) = [G,G]. If G is of class 2 , we have [#,t/]p = [#p,y] = 1 for every a?, y G G. So Φ(G) is elementary abelian. [] LEMMA 3.23. Let be a central extension with factor set z G H2(K). Set Z = iZp. Then: (a) d(K) < d(G) < d(K] + 1. Moreover, d(G] = d(K) + I iff the extension splits, i.e. z = O ιnH2(K). (b) if z ^ 0 in H2(K), then G is not d-maximal provided that z is decomposable (i.e. A COHOMOLOGICAL APPROACH TO THEORY OF GROUPS 581 z - x.y with x,y e H1^)), or z = βx with x € Hl(K). (c) for every subgroup L ofG, we have d(L) < d(L.Z), and d(L) — d(L.Z) iff Z C L. Proof, (a) It is clear that d(K) < d(G). By Prop. 1.4. we have d(G) < d(K) + 1, and d(G) = d(K) iff Z (jL Φ(G), or equivalently, z - 0. (b) Let L = Kerx C G. Then d(L/Z) = d(G/Z)-l = d(G)-l. Since Έtes(L/ZtK)z = 0, the extension 0 -> Zp -> L -> L/Z -> 1 splits. Hence d(L) = d(L/Z)+l = d(G). So G is not d-maximal. (c) If Z £ L, then L.Z = L x Z, so d(L.Z) = d(i) + 1. Hence d(L) < d(L.Z) and d(L) = cf(L.Z) iff Z C L. 0 (3.24) Let G be an almost special p-group given by the central extension 0 _* W -> G -> F -+ 0 2 with factor set z G H (V, W), where FT and V are respectively vector spaces of dimension ?τι, n over Zp. Let {ei, ...,em} be a fixed base of W. The factor set z is then of form z = (*ι,... ,z m ) with *i = pfa) G fT 2 (VO. If / G GL(W), it is obvious that G is isomorphic to the extension of W by V with factor set f*z. Set r(z) = rank(zι,...,z m ) in #2(F). By Theorem 2.1 and the fact that Kerlnf (H*(V) -> #2(G)) - (zι,...,* m ), we have LEMMA 3.25. With the notations o/(3.24), //ten Φ(G) = W tj^Kz) = m. Let {tίi, ...,iίn} be a base of Hl(V). Each Ui is consedered as element of Hl(G) via the inflation map. From Lemmas 3.23 c) and 3.25, we get the following lemma, which has been proved by Kahn in [13]. LEMMA 3.26. With the notations o/(3.24), let I < k < n be an integer and L = ΠίU Kerw*cG Then d(Lϊ ~ d(G) ~k + m ~ Γ(Z\L) mtfl Z\L~ Ίte8(L/W, V}z. Hence G is d-maximal iff m <k + r(z|χ,) for every 1 < k < n. (3.27) For convenience, with a given subgroup L — Πt =ι Keriίz of G as in Lemma 3.26 and X G H*(V), we write X1 = X\L = Res(L/VΓ, 7)X. With this notation, *x is then obtained from z by setting u\ = ... = Uk = 0, /??/ι — ... = βuk — 0. In the following three lemmas , G is assumed to be given as in (3.24) with r ( z ) — m. LEMMA 3.28. If there exist x\, ...,xm G Hl(V) not all equal to 0 such that x\z\ -f ... -f XmZm = 0 or XiZi + βz\ — 0, then G is not d-maximaί Proof. If xizi + ... + xmzm = 0, we can suppose that xΐy ...,a? m are linearly independent. So, for every i, we have Z{ G (a?i, ..., xm,βxι, ...,/?x m ), the ideal generated by ίci, ~>Xm,βxiί - ,β%m Hence r(z') = 0 with L = p|^:1 Kerxf. By Lemma 3.26, G is not d-maximal. If x\z\ -f ^i = 0, z\ is then of form z\ — βu + x\.v with i/, υ G Hl(V). We have then βzi + x\z\ — 0. For p > 2, this implies /?X!.D — xι./?υ -f xι./?ι« — 0, so v = λxi, with λ G 2p. For p = 2, xι[zι + (#ι + υ)t;] = 0, so 21 = v(xι + v). By Lemma 3.23 b), G 582 PHAM ANH MINH is not {/-maximal. [] LEMMA 3.29. If there exist xι,...,z m G Hl(V] not all equal to 0 such that X = xizi -i- ... + xmxm + βzm = 0, then G is not d-maximal provided that p > 2, or p = 2 and m — 2. Proof. By Lemma 3.28, we can assume that ra > 2. Since X = 0, 2m is of form 1 v Zm = /?K + ϋιZι + ... + ϋm*m with tl, V, G Jί ^)- Hence Y = Σ!ΪLl\fl( i) +zi\xi-βχi vi = 0. Consider the following cases: (a) p > 2: for every 1 < k < m, set H - Π^jfeKert;,-, then 0 = Y\H = [/?(tijb) + **]sfc - β(%k}.Vk Hence ^ G (a?ι,...,x m ) and zfc G (xi, ...,xm,βxι, ...,βxm). We have then r(z') — 0 with Z/ = Πί=ι Kerx 2 . By Lemma 3.26, G is not d-maximal. (b) p = 2 and m = 2: since 0 = Y = x\[vι(xι + vι) + zι] + ^2^2(^2 4- ^2) + ^2], we obtain zi = vι(vι H- #ι) + ^.x 2) ^2 = k.xi -f ^2(^2 + ^2) with k G ^Γ1^). So k(k + x\) = vι(υι + a?ι). This implies fc = ϋi or v\ + x\. Hence z^ is decomposable and G is not d-maximal by Lemma 3.23 b). [j LEMMA 3.30. I f G is d-maximal, then Inf(# 2 (V) -+ H2(G)) is surjectwe, provided that p > 2 or p = 2 αnrf m < 2. Proof. Obvious from Lemmas 3.28, 3.29 and (1.2). Proof of Theorem 3.31. The proof is evident from Lemma 3.30 and Theorem 2.1. Proof of Theorem 3.32. The proof follows from Theorem 3.18, Theorem 2.1 and Proposition 1.4. Proof of Theorem 3.33. Let G be the almost special 2-group given by the central extension with factor set z = ( z \ , Z ϊ , z z ) in which Z3 = y2 + a2 + b2 + ax + by, with {x,y,α,6} a fixed base of Hl(Z$). By Lemma 3.26 and by a direct verification, we can show that G is d-maximal. 3 Clearly d(G) = 4 and |Φ(G)| = 2 . Since βz3 = y^! H- zz 2 , following Proposition 1.1, there exists a non-zero element V G H2(G) such that V|^ = tί2 φ 0, so V|φ(^) ^ 0. Let Γ be the central extension with factor set V, then d(Γ) = 4. Assume that there exists a subgroup L of Γ such that d(L) > 4. By Lemma 3.23 c), we can suppose that Z1 = j^2 C />. Since L/Z1 is a A COHOMOLOGICAL APPROACH TO THEORY OF GROUPS 583 subgroup of G and G is d-maximal, d(L/Z'} < 3. By Lemma 3.23 a), d(L) > 4 implies the splitting of the extension 1 -* Z' -> L -> L/Z' -> 1 1 and d(L/Z') = 3,Z C L/Z . We have then V\L/Z' - 0. This contradicts the fact that V\z ^ 0. So Γ is cί-maximal. Since V\φ(G) ± 0, Φ(Γ) is not elementary abelian. By Lemma 3.2, Φ(Γ) £ Z(Γ). Hence Γ is of class > 3. For every n > 4, set G\ — — x Zς~ . I t is clear that Q\ is d-maximal of class > 3 and d(Q\) = \. The theorem follows. The following is straighforward from Corollary 2.4, Lemma 3.28 and 3.33. COROLLARY 3.34. For every p-group G, G ts d-maximal ι f f G / Φ l G for any i > 2. Furthermore, if G is d-maximal, then (Φ1G)P = Φ*+1G. is d-maximal, Remark 3.35. In [13], Kahn claimed that every d-maximal 2-group H with d(H) = 4 is of class < 2, by proving that if cl(H) = 3, [Φ(#), H] = ^2> then there exists an element x G H — Φ(H) such that x2 G Φ(#)2. This proof is not correct. 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