EXTRASPECIAL 2-GROUPS AND

mod-2 COHOMOLOGY LENGTHS

Pham Anh Minh

Department of Mathematics, College of Sciences, University of Hue Dai hoc Khoa hoc, Hue, Vietnam Email: paminh@dng.vnn.vn

Abstract. Let G_{0} (resp. G_{1}, G_{2}) be the extraspecial 2-group of real (resp. complex, quaternion) type of order 2^{2n+3} (resp. 2^{2n+2}, 2^{2n+1}). The purpose of this paper is to investigate the cohomology lengths chl(G_{i}) with i = 1, 2 (the case i = 0 was settled up by Yalçın). We prove that chl(G_{0}) ≥ chl(G_{1}) ≥ chl(G_{2}) > 2^{n}, and the equalities chl(G_{0}) = chl(G_{1}) = chl(G_{2}) hold for n ≤ 3. This disproves a conjecture of Yalçın telling that chl(G_{i}) = s(G_{i}), i = 1, 2, with s(G_{i}) the minimum cardinality of a representing set of G_{i}.

1. Statement of the main result

Let G be a p-group. Denote by H^{∗}(G) the mod-p cohomology of G. Suppose that G is not elementary abelian. The cohomology length of G, denoted by chl(G), is defined to be the smallest integer m such that there exist non-zero elements x_{1}, …, x_{m} of H^{1}(G) satisfying

\begin{aligned}x_{1} · . . . · x_{m} & = 0 for p = 2, \\ βx_{1} · . . . · βx_{m} & = 0 for p > 2, \end{aligned}

with β the Bockstein operation (the existence of such an integer follows from a classical theorem of Serre [S1]).

The following upper bounds for chl(G), in terms of k = dim_{ℤ/p} H^{1}(G), were known: p^{k} − 1 (Kroll [K]), \frac{p^{k}−1}{p−1} (Serre [S2]), (p + 1)p^{k−2} (Okuyama-Sasaki [OS]), (p + 1)p^{[ \frac{k}{2} ]−1} (Minh [M]), (p^{2} + p − 1)p^{[ \frac{k}{2} ]−2} for k ≥ 2 (Yalçın [Y]).

Note that the cohomology length is increasing on factor groups (i.e. chl(G) ≤ chl(K) for K a non-elementary abelian factor group of G). Since G is an extension of an extraspecial p-group, chl(G) is bounded above by cohomology lengths of extraspecial factor p-groups of G.

We are now interested in the cohomology lengths of extraspecial 2-groups. Without loss of generality, we assume that our extraspecial 2-groups have no proper direct factors. In [Y], Yalçın has determined the cohomology length of the extraspecial 2-group G_{0} of real type of order 2^{2n+3}. His argument is to calculate the minimum cardinality s(E) of a representing set of any extraspecial 2-group E (recall that S is a representing set of E if it includes at least one non-central element from each maximal elementary abelian subgroup of E). He showed that chl(G_{0}) = s(G_{0}) and got the following

Typeset by AMS-TEX

2000 Mathematics Subject Classification. 20J06.

Theorem 1 (Yalçın [Y]).

chl(G_{0}) = \{ \begin{matrix}2^{n} + 1 & for n ≤ 3, \\ 2^{n} + 2^{n−3} & for n ≥ 4. □ \end{matrix}

The purpose of this paper is to investigate the cohomology lengths of extraspecial 2-groups of remaining types. Let G_{1} (resp. G_{2}) be the extraspecial 2-group of order 2^{2n+2} (resp. 2^{2n+1}) of complex (resp. quaternion) type. In [Y, Propositions 6.2 and 6.3], if G_{2} ≅ Q̸_{8}, Yalçın also proved

s(G_{1}) = s(G_{2}) = \{ \begin{matrix}2^{n} + 1 & for n ≤ 1, \\ 2^{n} + 2^{n−2} & for n ≥ 2, \end{matrix}

and proposed a problem ([Y, Problem 7.1]) showing that chl(G_{i}) = s(G_{i}), 1 ≤ i ≤ 2.

The main result of our paper is the following

Theorem 2. (i) chl(G_{0}) ≥ chl(G_{1}) ≥ chl(G_{2}) > 2^{n}.

(ii) If n ≤ 3, then chl(G_{0}) = chl(G_{1}) = chl(G_{2}).

It turns out that chl(G_{i}) < s(G_{i}) for n ≥ 3, i = 1, 2. Hence, in general, the equalities chl(G_{i}) = s(G_{i}), 1 ≤ i ≤ 2, as conjectured by Yalçın [Y], do not hold.

However, we could not determine chl(G_{i}) for i = 1, 2 and n ≥ 4. Determination of cohomology lengths of such groups still remains as an open problem.

2. Proof of the main result

Denote by C_{4} the cyclic group of order 4, and by D_{8} (resp. Q_{8}) the dihedral (resp. quaternion) group of order 8. It is known that each G_{i}, 0 ≤ i ≤ 2, is a central product of the groups D_{8}, Q_{8} (and C_{4} for i = 1); namely

G_{i} = \{ \begin{matrix}\underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n+1 times} & for i = 0, \\ C_{4} ∗ \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n times} & for i = 1, \\ Q_{8} ∗ \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n−1 times} & for i = 2. \end{matrix}

Consider each G_{i+1}, 0 ≤ i ≤ 1, as a maximal subgroup of G_{i}. Let Z be the center of G_{0} and set V_{i} = G_{i}/Z, 0 ≤ i ≤ 2. V_{0} is then elementary abelian and contains V_{1}, V_{2} as subgroups. Let z_{0} ∈ H^{2}(V_{0}) be the cohomology class classifying the central extension

1 → Z → G_{0} → V_{0} → 0.

Via the inflation map, every element of H^{∗}(V_{i}) can be considered as an element of H^{∗}(G_{i}). Following [Q], there exists ξ_{n,0} ∈ H^{2^{n+1}}(G_{0}) which restricts non-trivially to H^{2^{n+1}}(Z). Set ξ_{n,i} = Res_{G_{i}}^{G_{0}}(ξ_{n,0}), z_{i} = Res_{V_{i}}^{V_{0}}(z_{0}), 1 ≤ i ≤ 2. For convenience, write ξ_{n,i} = ξ_{i}, if no confusion can arise. Let Sq^{j}, j ≥ 0, be the Steenrod squares. We have

Theorem 3 (Quillen [Q]). For 0 ≤ i ≤ 2,

H^{∗}(G_{i}) = ℤ/2[V_{i}^{∗}]/(z_{i}, Sq^{1}z_{i}, Sq^{2}Sq^{1}z_{i}, …, Sq^{2^{n−1}} … Sq^{1}z_{i}) ⊗ ℤ/2[ξ_{i}]  □

We shall need Lemma 1. Let K_{i} be a maximal subgroup of G_{i}, 0 ≤ i ≤ 2. Then

K_{i} ≅ \{ \begin{matrix}G_{1} or \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n times} ×ℤ/2 & for i = 0, \\ G_{2}, \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n times} or C_{4} ∗ \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n−1 times} ×ℤ/2 & for i = 1, \\ Q_{8} ∗ \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n−2 times} ×ℤ/2 or C_{4} ∗ \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n−1 times} & for i = 2. \end{matrix}

Proof. Since Z ⊂ K_{i} and K_{i}/Z is elementary abelian, K_{i} is either an extraspecial 2-group, or a direct product of an extraspecial 2-group and an elementary abelian group. If K_{i} is extraspecial, as it is of order \frac{|G_{i}|}{2}, it is isomorphic to G_{1} for i = 0, G_{2} or \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n times} for i = 1, and C_{4} ∗ \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n−1 times} for i = 2. Suppose that

K_{i} is a direct product of an extraspecial 2-group L and an elementary abelian group A. The subgroups L and A can be chosen such that there exists a ∈ G_{i}\K_{i} satisfying [L, a] = 1 and C_{G}(a) ∩ A is maximal in A. Since G_{i} = ⟨K_{i}, a⟩, it follows that C_{G}(a) ∩ A is a direct factor of G_{i}. So C_{G}(a) ∩ A = 1, as G_{i} has no proper direct factor. Therefore A = ℤ/2. L is then extraspecial of order \frac{|G_{i}|}{4}. So

L ≅ \underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n times} for i = 0, C_{4} ∗\underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n−1 times} for i = 1, Q_{8} ∗\underbrace{D_{8} ∗ ⋯ ∗ D_{8}}_{n−2 times} for i = 2.

The lemma is proved □

Note that, for every non-zero element x of H^{∗}(G_{i}), by setting L_{i} = ker x, we have the exact sequence (which is known to be the Gysin sequence of the 0-sphere bundle BL_{i} → BG_{i}):

(∗)  ⋯ → H^{∗−1}(G_{i}) →^{·x} H^{∗}(G_{i}) →^{Res_{L_{i}}^{G_{i}}} H^{∗}(L_{i}) →^{tr_{G_{i}}^{L_{i}}} H^{∗}(G_{i}) → …

with Res_{L_{i}}^{G_{i}} (resp. tr_{G_{i}}^{L_{i}}) the restriction (resp. transfer) map from H^{∗}(G_{i}) (resp. H^{∗}(L_{i})) to H^{∗}(L_{i}) (resp. H^{∗}(G_{i})).

Denote by Inf_{G_{i}}^{V_{i}} (resp. Inf_{L_{i}}^{L_{i}/Z}) the inflation map from H^{∗}(V_{i}) (resp. H^{∗}(L_{i}/Z)) to H^{∗}(G_{i}) (resp. H^{∗}(L_{i})). We also have the commutative diagram

(∗∗)  H^{∗}(V_{i}) ⟶^{Inf_{G_{i}}^{V_{i}}} H^{∗}(G_{i}) ^{Res}_{L_{i}/Z}^{V_{i}} ↓  ↓^{Res}_{L_{i}}^{G_{i}} H^{∗}(L_{i}/Z) ⟶_{Inf_{L_{i}}^{L_{i}/Z}} H^{∗}(L_{i}) .

Using Lemma 1, we have

Lemma 2. Let K_{i} be a maximal subgroup of G_{i}, 0 ≤ i ≤ 2.

(i) If K_{i} ≅ G_{i+1}, then Res_{K_{i}}^{G_{i}} is surjective and tr_{G_{i}}^{K_{i}} is the zero map.

(ii) If K_{i} is not isomorphic to G_{i+1}, then Im tr_{G_{i}}^{K_{i}} is generated, as an ideal, by a cohomology class of degree 2^{n}.

Proof. As V_{i} is elementary abelian, it contains K_{i}/Z as a direct factor. So Res_{K_{i}/Z}^{V_{i}} is surjective.

(i) The surjectivity of Res_{K_{i}}^{G_{i}} follows from Theorem 3, the definition of the ξ_{i}’s and the commutative diagram (∗∗). Since Res_{K_{i}}^{G_{i}} is surjective, the exact sequence (∗) implies ker tr_{G_{i}}^{K_{i}} = Im Res_{K_{i}}^{G_{i}} = H^{∗}(K_{i}). Hence tr_{G_{i}}^{K_{i}} is the zero map. (ii) By Künneth formula, Theorem 3 and the commutative diagram (∗∗), Im tr_{G_{i}}^{K_{i}} is generated, as an ideal, by

tr_{G_{i}}^{K_{i}}(ξ_{n−1,i}) for i = 0,
tr_{G_{i}}^{K_{i}}(ξ_{n−1,i}) or tr_{G_{i}}^{K_{i}}(ξ_{n−1,0}) for i = 1,
tr_{G_{i}}^{K_{i}}(ξ_{n−1,2}) or tr_{G_{i}}^{K_{i}}(ξ_{n−1,1}) for i = 2

which is a cohomology class of degree 2^{n}.

The lemma is proved. □

We shall use the following notation. Set m_{i} = chl(G_{i}), 0 ≤ i ≤ 2. Pick non-zero

elements x_{j}^{(i)} ∈ H^{1}(G_{i}), 1 ≤ j ≤ m_{i}, satisfying 0 = ∏_{j=1}^{m_{i}} x_{j}^{(i)} ∈ H^{m_{i}}(G_{i}).

Lemma 3. Let L_{i} be a maximal subgroup of G_{i}, 0 ≤ i ≤ 1, which is isomorphic to G_{i+1}. Then Res_{L_{i}}^{G_{i}}(x_{j}^{(i)}) ≠ 0, for 1 ≤ j ≤ m_{i}.

Proof. Suppose that Res_{L_{i}}^{G_{i}}(x_{k}^{(i)}) = 0 for some k with 0 ≤ i ≤ 1, 1 ≤ k ≤ m_{i}. So L_{i} = ker x_{k}^{(i)}. It follows from the definition of the cohomology length and the exact

sequence (∗) that 0 ≠ _{∏_{j=1}^{m_{i}} ; j≠k}x_{j}^{(i)} ∈ Im tr_{G_{i}}^{L_{i}}, which contradicts Lemma 2 (i). Therefore

Res_{L_{i}}^{G_{i}}(x_{j}^{(i)}) ≠ 0, 1 ≤ j ≤ m_{i}. □

Proof of Theorem 2. (i) Set L_{i} = ker x_{1}^{(i)}. As x_{1}^{(i)} ≠ 0 and Res_{L_{i}}^{G_{i}}(x_{1}^{(i)}) = 0, it follows that L_{i} is a maximal subgroup of G_{i} which is not isomorphic to G_{i+1}, by Lemma 3. By the definition of the cohomology length and by the exact sequence

(∗), 0 ≠ ∏_{j=2}^{m_{i}} x_{j}^{(i)} belongs to Im tr_{G_{i}}^{L_{i}}. By Lemma 2(ii), m_{i} − 1 = deg(∏_{j=2}^{m_{i}} x_{j}^{(i)}) ≥ 2^{n}.

So m_{i} ≥ 2^{n} + 1 > 2^{n}.

Set t_{j}^{(i)} = Res_{G_{i+1}}^{G_{i}}(x_{j}^{(i)}), 1 ≤ j ≤ m_{i}. By Lemma 3, 0 ≠ t_{j}^{(i)} ∈ H^{1}(G_{i+1}), 1 ≤ j ≤

m_{i}. As ∏_{j=1}^{m_{i}} t_{j}^{(i)} = Res_{G_{i+1}}^{G_{i}}(∏_{j=1}^{m_{i}} x_{j}^{(i)}) = Res_{G_{i+1}}^{G_{i}}(0) = 0, it follows from the definition

of the cohomology length that m_{i+1} ≤ m_{i}. Therefore chl(G_{0}) ≥ chl(G_{1}) ≥ chl(G_{2}).

(ii) Suppose now that n ≤ 3. It follows from Theorem 1 and what we just proved that 2^{n} < chl(G_{2}) ≤ chl(G_{1}) ≤ chl(G_{0}) = 2^{n} + 1. So chl(G_{0}) = chl(G_{1}) = chl(G_{2}) = 2^{n} + 1.

The theorem is proved. □

References

[K] O. Kroll, A representation theoretical proof of a theorem of Serre, Århus Preprint, Maj 1986.

[M] P. A. Minh, Serre’s theorem on the cohomology algebra of a p-group, Bull. London Math. Soc. 30 (1998), 518-520.

[OS] T. Okuyama and H. Sasaki, Evens’ norm maps and Serre’s theorem on the cohomology algebra of a p-group, Arch. Math 54 (1990), 331-339.

[Q] D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197-212.

[S1] J. P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413-420.

[S2] , Une relation dans la cohomologie des p-groupes, C. R. Acad. Sci. Paris 304 (1987), 587-590.

[Y] E. Yalçın, Set covering and Serre’s theorem on the cohomology algebra of a p-group, J. Alg. 245 (2001), 50-67.