COHOMOLOGY OF GROUPS WITH

METACYCLIC SYLOW p-SUBGROUPS

Jill Dietz, John Martino, and Stewart Priddy

1. Introduction and Statement of Results.

Let P be a non-abelian metacyclic p-group of odd order and G be a finite group with P as a Sylow p-subgroup. In this note we classify all possible mod-p cohomology algebras H^{∗}(G) and determine complete p-local stable splittings for the classifying spaces BG. Much of the topological part of this work was done by the first author in [D]; recent results on Swan groups [MP] have enabled us to show that in all cases H^{∗}(G) is given by a ring of invariants. Similar but less complete information for metacyclic 2-groups was obtained in [D1, MP2, M].

A metacyclic p-group is a p-group P which is an extension of a cyclic group by a cyclic group. Following [D] we say that P is split if P can be expressed by some split extension. We recall that up to isomorphism any non-abelian metacyclic p-group can be expressed as

P = P (p^{m}, p^{n}, p^{l} + 1, p^{q}) = ⟨x, y | x^{p^{m}} = 1, y^{p^{n}} = x^{p^{q}}, yxy^{−1} = x^{p^{l}+1}⟩

for positive integers m, n, l, q satisfying l, q ≤ m, (p^{l} +1)^{p^{n}} ≡ 1 mod p^{m}, (p^{l} +1)p^{q} ≡ p^{q} mod p^{m}, n + l ≥ m and q + l ≥ m. In these terms P splits unless m ≠ q and l < q < n [D, Thm. 3.1].

Let W_{G}(P) = N_{G}(P)/P · C_{G}(P) then W_{G}(P) ≤ Out(P). If P is split then Out(P) ≅ O_{p}Out(P) ⋊ Z/(p − 1) where O_{p}Out(P) is a Sylow p-subgroup [D;Prop. 3.2]. Therefore W_{G}(P) = Z/d where d is a divisor of p − 1. If P is non-split Out(P) is a p-group and so W_{G}(P) = 1. We denote by F_{p}[·] and E[·] the polynomial and exterior algebras over F_{p}.

Theorem 1.1. As an algebra, H^{∗}(G) has one of the following forms:

(1) If P is split and l ≠ m − n then

H^{∗}(G) ≅ H^{∗}(P)^{W_{G}(P)} = F_{p}[u_{d}, v] ⊗ E[a_{d}, b]

where |u_{d}| = 2d, |v| = 2, |a_{d}| = 2d − 1, |b| = 1.

Typeset by AMS-TEX

The third author is partially supported by NSF Grant DMS-9400235

(2) If P is split and l = m − n then

H^{∗}(G) ≅ H^{∗}(P)^{W_{G}(P)} = F_{p}[v, z] ⊗ E[b, α_{2i−1}, i = 1, .., p]/R

where the relations R are given by

α_{2i−1}α_{2j−1} = 0 1 ≤ i, j ≤ p
α_{2i−1}v = 0 1 ≤ i ≤ p − 1

and |b| = 1, |v| = 2, |z| = 2p, |α_{2i−1}| = 2i−1+2pd(i) where 0 ≤ d(i) < d is the residue of −i mod d.

(3) If P is non-split then H^{∗}(G) = H^{∗}(P) is isomorphic to the algebra of (1) with d = 1 if m = l + q and to that of (2) with d = 1 if m < l + q.

Generators for these cohomology groups are specified explicitly in the proof.

Remark 1.2. Groups exemplifying the cases above are easily given by G = P ⋊Z/d.

R. Lyons has suggested other, more natural examples, which occur as automorphism groups of Chevalley groups. For example, let F_{q} be a finite field of characteristic different from p such that the Sylow p-subgroup of P SL_{2}(F_{q}) has order p, i.e., q^{2}−1 is divisible by p but not by p^{2}. Then the Sylow p-subgroup of H = P SL_{2}(F_{q^{p}}) is cyclic of order p^{2}. Let ϕ be the Frobenius automorphism of F_{q^{p}} of order p. Then it is easy to see that ϕ fixes a cyclic subgroup C ≤ H of order p^{q} + 1 which contains one such Sylow p-subgroup. Thus G = Aut(H) = P SL_{2}(F_{q}) ⋊ Z/p⟨ϕ⟩ has P = M_{3}(p) as a Sylow p-subgroup. Furthermore N_{H}(C) is a dihedral group [Hu; II,8.4 Satz] containing the permutation matrix of order two. Since this matrix is fixed by ϕ, W_{G}(P) = Z/2 and H^{∗}(G) is of type (2) in Theorem 1.1 with d = 2.

The group cohomology of a group G is the cohomology of the classifying space BG of G. The space BG is stably homotopy equivalent to a wedge product of indecomposable spectra,

BG ≃ X_{1} ∨ X_{2} ∨ ⋯ ∨ X_{n}.

A complete stable decomposition of BG is a splitting into indecomposable spectra. The decomposition is unique up to stable homotopy type and ordering. If G is a p-group then all of these spectra are p-local. Otherwise, if P is a Sylow p-subgroup of G then a simple transfer argument shows the p-localization of BG is a stable summand of BP,

BP ≃ BG_{p} ∨ Y,

where BG_{p} is the p-localization of BG. Hence BG_{p} consists of some, but possibly not all of the summands of BP. Note H^{∗}(BG_{p}; F_{p}) = H^{∗}(BG; F_{p}).

Each indecomposable spectrum X of BP corresponds up to conjugacy to a primitive idempotent e in the ring of stable self-maps {BP, BP }. The spectrum X is the infinite mapping telescope or homotopy colimit of e,

X ≃ eBP = T el(BP ⟶^{e} BP) = hocolim(BP ⟶^{e} BP ⟶^{e} ⋯).

For more information see either [BF] or [MP1].

For the remainder of the paper all spectra are localized at the prime p. If P is a Swan group then BG ≃ BN_{G}(P) ≃ B(P ⋊ W_{G}(P)). Thus determining the stable homotopy type of BG involves determining which summands have their cohomology left invariant by the action of the Weyl group of G.

Z_{p}Out(P) ⊆ {BP, BP } is a subring, in fact a retract. Therefore, certain indecomposable summands of BP correspond to simple modules of the outer automorphism group Out(P). A summand corresponding to a simple Out(P)-module is said to originate in BP. A summand originating in BP does not occur as the summand of the classifying space of any proper subgroup of P.

In this paragraph we introduce some notation for Theorem 1.3 below. L(2, k) originates in B(Z/p×Z/p) and corresponds to St⊗(det)^{k} where St is the Steinberg module for F_{p}GL_{2}(F_{p}) and det is the determinant module. It is well known that the group ring F_{p}[Z/(p−1)] has a complete set of orthogonal primitive idempotents e_{0}, ..., e_{p−2} [D]. Lifting these idempotents to Z_{p}[Z/(p − 1)] determines a complete stable splitting of

BZ/p^{n} ≃ ⋁_{i=0}^{p−2} L(1, n, i),

where L(1, n, i) originates in BZ/p^{n}. For more information on these summands see [HK].

If P is a split metacyclic p-group then since W_{G}(P) is a p′-group we have W_{G}(P) ≤ Z/(p − 1). Thus the primitive idempotents e_{0}, ..., e_{p−2} above determine a stable splitting of BG. If P is non-split then BP is stably indecomposable [D; Thm. 1.3].

Among the split metacyclic groups there is one which plays a special role, the extra-special modular group M_{3}(p) = P (p^{2}, p, p + 1, 1). It is characterized by its order and exponent which are p^{3} and p^{2} respectively.

Theorem 1.3.

(1) If P is split and P ≠ M_{3}(p) then

e_{0}BP = X_{0} ∨ B(Z/p^{n}), e_{i}BP = X_{i}, l = m − n, 1 ≤ i ≤ p − 2. e_{0}BP = X_{0}∨B(Z/p^{n})∨L(1, n, 0), e_{i}BP = X_{i}∨L(1, n, i), l ≠ m−n, 1 ≤ i ≤ p−2.

(2) If P = M_{3}(p) then

e_{0}BP = X_{0} ∨ ⋁_{i=0}^{p−2} L(2, i) ∨ L(1, 1, i), e_{i}BP = X_{i} 1 ≤ i ≤ p − 2

where X_{i} originates in BP.

(3) In both cases this yields a complete stable decomposition of BP and

BG ≃ ⋁_{i=0}^{(p−1/d)−1} e_{id}BP.

Corollary 1.4. Localized at p, the complete stable decomposition of BG is given by one of the following:

(1) If P is non-split then BG ≃ BP.

(2) If P is split and P ≠ M_{3}(p) then

BG ≃ ⋁_{i=0}^{(p−1/d)−1} X_{id} ∨ ⋁_{j=0}^{p−2} L(1, n, j), l = m − n.

BG ≃ ⋁_{i=0}^{(p−1/d)−1} X_{id} ∨ L(1, n, id) ∨ ⋁_{j=0}^{p−2} L(1, n, j), l ≠ m − n.

(3) If P = M_{3}(p) then

BG ≃ ⋁_{i=0}^{(p−1/d)−1} X_{id} ∨ ⋁_{i=0}^{p−2} L(2, i) ∨ L(1, 1, i).

If P is split and P ≠ M_{3}(p) then Theorem 1.3 and Corollary 1.4 were proved by the first author [D; Thm. 1.3].

Throughout we assume P is a non-abelian metacyclic p-group and p is an odd prime. All cohomology is taken with simple coefficients in F_{p} and all spaces are considered stably, localized at p.

2. Proofs.

The classifying space BP is indecomposable if and only if P is non-split [D; Thm. 1.1]. Thus, since BG is a summand in BP, we have BG ≃ BP and H^{∗}(G) ≅ H^{∗}(P). In this case the cohomology algebras are given by [Hb; Thm. B if m < l + q and Thm. E if m = l + q]. This completes the proof of both theorems for the non-split case.

Before turning to the proof of Theorem 1.1 we recall the notion of a Swan group [MP]. A p-group P is called a Swan group if the cohomology of any group G with P as a Sylow p-subgroup is given by its invariants, i.e.,

res : H^{∗}(G; F_{p}) ⟶^{≅} H^{∗}(P; F_{p})^{W_{G}(P)}.

The following result of Dietz and Glauberman [MP] is fundamental to our classification.

Theorem 2.1. If P is a metacyclic group of odd order then P is a Swan group.

Proof of Theorem 1.1 for P split.

Let Φ(P) = ⟨x^{p}, y^{p}⟩ be the Frattini subgroup. Then P/Φ(P) ≅ Z/p × Z/p = ⟨\overline{x}, \overline{y}⟩. Thus quotienting by Φ(P) induces a homomorphism Out(P) ⟶^{π} Aut(P/Φ(P)) = GL_{2}(F_{p}). By [D; Prop. 3.2] if P is split then Out(P) ≅ O_{p}Out(P) ⋊ Z/(p − 1); moreover any automorphism of P of p′ order gives rise to a matrix of the form

(1)  ( \begin{matrix}∗ & 0 \\ 0 & 1 \end{matrix} ) ∈ Im(π)

Now consider the extension

1 → N → P → K → 1

where N ≅ Z/p^{m} = ⟨x⟩ and K ≅ Z/p^{n} = ⟨\overline{y}⟩. The Lyndon-Hochschild-Serre spectral sequence has E_{2} = F_{p}[u, v]⊗E[a, b] where H^{∗}(N) = F_{p}[u]⊗E[a], H^{∗}(K) = F_{p}[v] ⊗ E[b], |a| = |b| = 1, |u| = |v| = 2. Explicitly a, b are given as canonical homomorphisms dual to x, y respectively and u = β_{n}(a), v = β_{m}(b) are their respective Bocksteins.

If l ≠ m − n then this spectral sequence collapses at E_{2} [Dh;Thm.1]. Since P is a Swan group we need only compute invariants. Let ζ be a generator of Z/(p − 1) ≤ Out(P) so that γ = ζ^{p−1/d} generates W_{G}(P) = Z/d. By (1) γ^{∗}(a) = c · a where c^{d} = 1 is a primitive d-th root of unity; γ^{∗}(u) = c·u by application of the Bockstein. Similarly γ^{∗} is trivial on v, b. Computing we find γ^{∗}(u^{k}v^{l}a^{ϵ}b^{δ}) = u^{k}v^{l}a^{ϵ}b^{δ} iff k + ϵ ≡ 0 mod d. Theorem 1.1 (1) follows with u_{d} = u^{d}, a_{d} = u^{d−1}a.

If l = m − n the spectral sequence collapses at E_{3} [Dh;Thm.2] and we have E_{3} = F_{p}[z, v] ⊗ E[b, ξ_{2i−1}, i = 1, ..., p]/R where z = u^{p}, ξ_{2i−1} = au^{i−1}. Relations are given by

ξ_{2i−1}ξ_{2j−1} = 0, 1 ≤ i, j ≤ p,
ξ_{2i−1}v = 0, 1 ≤ i ≤ p − 1.

In this case, γ^{∗}(z) = c · z, γ^{∗}(ξ_{2i−1}) = c^{i} · ξ_{2i−1}. For d > 1 we have invariants z^{d} and α_{2i−1} = ξ_{2i−1}z^{d(i)}, where 0 ≤ d(i) < d is the residue of −i mod d. The result follows. For d = 1, H^{∗}(G) = H^{∗}(P) and the result holds setting α_{2i−1} = ξ_{2i−1} since d(i) = 0 in this case. □

Proof of Theorem 1.3 for P = M_{3}(p). In this case we have [D; Thm. 1.1 (3)]

BP ≃ ⋁_{i=0}^{p−2} X_{i} ∨ ⋁_{k=0}^{p−2} L(2, k) ∨ ⋁_{k=0}^{p−2} L(1, 1, k)

Since P is a Swan group we may assume G = N_{G}(P) and C_{G}(P) < P i.e., P ◁ G and G = P ⋊ C where C ≤ Z/(p − 1). From (1) in the proof of Theorem 1.1 it is clear the subgroup ⟨x⟩ ⋊ C is normal in G = P ⋊ C. Therefore, Z/p⟨y⟩ is a retract of G; hence, BZ/p is a summand of BG for every G with M_{3}(p) as a Sylow p-subgroup. Thus

BZ/p = ⋁_{k=0}^{p−2} L(1, 1, k)

is a summand of e_{0}BP = B(P ⋊ Z/(p − 1)).

We are reduced to showing L(2, k) is a summand of e_{0}BP. Let Q = ⟨x^{p}, y⟩ ≅ Z/p × Z/p. Since L(2, k) corresponds to the simple F_{p}Out(Q) module M_{k} = St ⊗ (det)^{k} we can use the criterion developed in [MP1]. That is, since Q is not a retract of P and C_{P} (Q) = Q, we must show

\overline{N_{G}(Q)/Q} · M_{k} ≠ 0

where \overline{H} = ∑ h summed over h ∈ H ≤ F_{p}(H). Since C_{G}(Q)/Q is a p′-group, this is equivalent to

\overline{N_{G}(Q)/C_{G}(Q)} · M_{k} ≠ 0

where N_{G}(Q)/C_{G}(Q) ≤ GL_{2}(F_{p}). An explicit description of the Steinberg module St may be given as follows: St = ⟨u^{p−1}, u^{p−2}v, … uv^{p−2}, v^{p−1}⟩ is the F_{p}-module of polynomials in indeterminates u, v of homogeneous degree p − 1 with GL_{2}(F_{p}) acting on ⟨u, v⟩ in the standard way [G]. Furthermore

\overline{N_{G}(Q)/C_{G}(Q)} = \overline{N_{G}(Q)/P C_{G}(Q)} · \overline{P/Q}

According to [D;Prop. 4.6 & proof]

\overline{P/Q} · M_{k} = ⟨v^{p−1}⟩

Since N_{G}(Q)/C_{G}(Q) is a p′-group normalizing P/Q we may assume it is isomorphic to a subgroup of the Borel subgroup of upper triangular matrices, i.e., the matrices of the form

w = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )

Thus w(v^{p−1}) = b^{p−1}v^{p−1} = v^{p−1} and so \overline{N_{G}(Q)/P C_{G}(Q)} · v^{p−1} ≠ 0. □

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Department of Mathematics, University of Washington, Seattle, WA 98195 E-mail address: dietz@math.washington.edu

Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008 E-mail address: martino@wmich.edu

Department of Mathematics, Northwestern University, Evanston, IL 60208 E-mail address: s priddy@math.nwu.edu