On the p-modular cohomology algebra of a finite p-group and a theorem of Serre ^{∗}

L’ algèbre de cohomologie p-modulaire d’un p-groupe fini

Koen Thas^{†}

Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium

                      kthas@cage.UGent.be

Abstract

We solve a problem posed by E. Yalçin on the cohomology length of a pgroup P, by providing bounds for the group theoretical invariant s(P) when p > 2. These bounds improve the known bounds on the cohomology length of p-groups for odd p.

Résumé. On obtient une borne pour la longueur cohomologique d’un p-groupe fini, p > 2, résolvant ainsi un problème posé par E. Yalçin.

1 Version Française abrégée

Soient p un nombre premier et P un p-groupe fini. Notons

H^{∗}(P) = H^{∗}(P, 𝔽_{p}) = ⨁_{i=0}^{∞} H^{i}(P, 𝔽_{p})

l’algèbre de cohomologie de P à coefficients dans 𝔽_{p}, et chl(P) la longueur cohomologique de P (voir section 2). Je me propose de démontrer le théorème suivant, qui résout un problème posé par E. Yalçin:

∗MSC 2000: 20J06 †The author is a Postdoctoral Fellow of the Fund for Scientific Research — Flanders (Belgium).

Theorem 1.1 Supposons que |P | = p^{2n+1} pour p > 2 et n ≥ 3. Si P est un p-groupe extra-spécial de type (d), on a

p^{n−2}(p^{2} + (\sqrt{2} − 1) − 5/2) + 1 ≤ chl(P) ≤ p^{n−2}(p^{2} + p − 1).

ou P_{2} est un p-groupe extra-spécial de type (d) avec |P_{2}| = p^{5}.

2 Introduction and notation

Throughout this paper, for a finite p-group P,

H^{∗}(P) = H^{∗}(P, 𝔽_{p}) = ⨁_{i=0}^{∞} H^{i}(P, 𝔽_{p})

will denote the p-modular cohomology algebra of P. A theorem of J.-P. Serre [7] states that if P is a p-group which is not elementary abelian, then there exist non-zero elements u_{1}, u_{2}, …, u_{m} ∈ H^{1}(P, 𝔽_{p}) such that

(∗) ∏_{i=1}^{m} u_{i} = 0 if p = 2 and ∏_{i=1}^{m} β(u_{i}) = 0 if p > 2,

where β is the Bockstein homomorphism. The smallest integer m such that relation (*) is satisfied is referred to as the cohomology length of P, and is denoted by chl(P) throughout. Several papers on the calculation of the cohomology length have appeared; see, for instance, O. Kroll [3], J.-P. Serre [8], T. Okuyama and H. Sasaki [6], P. A. Minh [5] and E. Yalçin [10]. Suppose P is a p-group which is not p-central (not all elements of order p belong to the center). Define a representing set S of P as a subset that includes at least one non-central element from each maximal elementary abelian subgroup of P. Then define s(P) as the minimum cardinality of a representing set in P.

Theorem 2.1 (E. Yalçin [10]) If P is an extra-special p-group which is not pcentral, then chl(P) ≤ s(P). Moreover, if P has a self-centralizing maximal elementary abelian subgroup, then equality holds.

Theorem 2.1 was applied in [10] to prove the following theorem, which yields the best known bound for chl(P):

Theorem 2.2 (E. Yalçin [10]) If P is a p-group and k = dim_{𝔽_{p}}H^{1}(P, 𝔽_{p}), then

chl(P) ≤ p + 1

if k ≤ 3, and for k > 3 we have

chl(P) ≤ (p^{2} + p − 1)p^{⌊k/2⌋−2}.

In this note, we give an inductive bound when p is an odd prime which yields a new lower bound. As such, we solve Problem 7.2 of E. Yalçin [10]. The precise statement of the main result will be made in the next section.

Acknowledgment. The author wishes to thank E. Yalçin for several helpful remarks on a first draft of the paper.

3 Extra-special p-groups and statement of the main result

Let P be an extra-special p-group, which in this note we define by the following group extension:

1 ↦ ℤ/p ↦ P ↦ V ↦ 1,

V being a vector space over 𝔽_{p}. Put k = dim_{𝔽_{p}}V. If P ≅ P^{∗} × ℤ/p for some subgroup P^{∗} ⊂ P, then chl(P) = chl(P^{∗}) and s(P) = s(P^{∗}). Without loss of generality we suppose that P is not of this form, that is, P has no proper direct factors. Then, if P is represented by the extension class [α] ∈ H^{1}(V, 𝔽_{p}), there exists a basis such that [α] is of one of the following forms (cf. P. A. Minh [5]):

for p = 2 and k = 2n,  (a) X_{1}Y_{1} + X_{2}Y_{2} + … + X_{n}Y_{n} or
for p = 2 and k = 2n,  (b) X_{1}^{2} + Y_{1}^{2} + X_{1}Y_{1} + X_{2}Y_{2} + … + X_{n}Y_{n};
for p = 2 and k = 2n + 1, (c) X_{0}^{2} + X_{1}Y_{1} + X_{2}Y_{2} + … + X_{n}Y_{n};
for p > 2 and k = 2n,  (d) X_{1}Y_{1} + X_{2}Y_{2} + … + X_{n}Y_{n} or
for p > 2 and k = 2n,  (e) β(X_{1}) + X_{1}Y_{1} + X_{2}Y_{2} + … + X_{n}Y_{n};
for p > 2 and k = 2n + 1, (f) β(X_{0}) + X_{1}Y_{1} + X_{2}Y_{2} + … + X_{n}Y_{n}.

When P is an extra-special group of type (e) or (f), it is well-known that chl(P) ≤ p. In Cases (a) and (d), C_{P} (E) = E for any maximal elementary abelian subgroup E ≤ P, so that equality holds in Theorem 2.1. For Case (a), E. Yalçin obtained the best possible bound in [10]. Theorem 2.2 represents a general bound which is valid for all cases. In that same paper, Problem 7.2 asks for a calculation of s(P) = chl(P) in terms of n and p for groups of type (d). This calculation is the objective of the present note, so as to obtain at the same time a bound for chl(P) of such a group P, and more generally, of any p-group for odd p.

Theorem 3.1 Let P be an extra-special p-group of order p^{2n+1} where p is odd. If P is of type (d), we have p^{n−2}(p^{2} + (\sqrt{2} − 1) − 5/2) + 1 ≤ chl(P) ≤ p^{n−2}(p^{2} + p − 1). Moreover,

p^{n−2}(chl(P_{2}) − 1) + 1 ≤ chl(P) ≤ p^{n−2} · chl(P_{2}),

where P_{2} is an extra-special p-group of type (d) with order p^{5}.

In the rest of this note, we will only consider extra-special groups of type (d); if P is a p-group, p odd, which is not elementary abelian and k = dim_{𝔽_{p}}(H^{1}(P, 𝔽_{p})) ∈ {2n, 2n + 1}, then P has a factor group P_{n} isomorphic to some group of type (d),

(e) or (f). So chl(P) ≤ chl(P_{n}).

4 Proof of the main result

Suppose P = P_{n} is a group of type (d), and note that |P_{n}| = p^{2n+1}. Suppose W = W(2n − 1, p) is the variety in the (2n − 1)-dimensional projective space PG(2n − 1, p) over 𝔽_{p} which is determined by the bilinear alternating form induced by the quadratic form displayed in (d) of the previous section. So W is a “nonsingular symplectic polar space”. Define s(W) as the minimal cardinality of a set of points of PG(2n − 1, p) which meets every maximal totally isotropic subspace (“generator”) of W. Then it holds that s(P_{n}) = s(W) [11]. Note that the Witt index of W is n, so that (n − 1) is the dimension of a generator of W. An easy counting argument^{1} shows that the number of points of such a set is at least p^{n} + 1 [9], and in case of equality, one speaks of an “ovoid” of W(2n − 1, p). More generally, if B is a point set of W(2n − 1, p) meeting each generator, call it a blocking set.

Theorem 4.1 (See, e.g., the survey paper [9] for (i) and [2] for (ii))

(i) W(2m + 1, p) has no ovoids for m ≥ 1.

(ii) s(W(3, p)) ≥ p^{2} + (\sqrt{2} − 1)p − 3/2.

We need a good bound for the size of a blocking set of symplectic polar spaces, which we will try to obtain now. Let W(2r − 1, p) ⊂ W(2n − 1, p), where we assume r ≥ 2 and n ≥ 3. Suppose η is the symplectic polarity defined by W(2n − 1, p). Now let π ⊂ W(2n − 1, p) be a projective subspace of PG(2n − 1, p) of dimension n − r − 1, such that W(2r − 1, p) ⊂ π^{η} (W(2r − 1, p) has no point in common with π). Suppose B is a blocking set of W(2r − 1, p). Define B^{∗} as the set of points of W(2n − 1, p) which are on lines that contain a point of B and one of π, but not contained in π (B^{∗} is a “truncated cone” with base π and vertex B). Then one observes two facts:

(i) |B^{∗}| = \frac{p^{n−r}−1}{p−1}(p − 1)|B| + |B| = |B|p^{n−r}; (ii) B^{∗} contains at least one point of any generator of W(2n − 1, p).

1Let S be a set of points of W(2n − 1, p) meeting every generator. Count in two ways the number of pairs (p, π), where p ∈ S, π is a generator and p ∈ π. Then |S|·(number of generators containing x) ≥ (total number of generators)·1.

Now put 2r −1 = 3, and consider a point x of W(3, p). Let y be a point of W(3, p) which is not collinear with x on W(3, p). For any point z of W(3, p), denote by z^{⊥} the set of points which are collinear with z on W(3, p) (including z). Also, if A is a point set of W(3, p), write A^{⊥} for ∩_{a∈A}a^{⊥}, and A^{⊥⊥} for (A^{⊥})^{⊥}. Then clearly

B = ((x^{⊥} \ {x, y}^{⊥}) ∪ {x, y}^{⊥⊥}) \ {x}

is a blocking set of W(3, p) of size p^{2} + p − 1. So we get

|B^{∗}| = p^{n−2}(p^{2} + p − 1),

and hence s(W(2n−1, p)) is at most p^{n−2} ·s(W(3, p)). (Note that as thus we have obtained an alternative proof of a result of [10].)

Now suppose that for k < n, k ∈ ℕ \ {0, 1}, s(W(2k − 1, p)) ≥ p^{k−2}(s(W(3, p)) − 1) + 1. We will use this induction hypothesis to show that the inequality also holds for k = n. The following argument was first made by K. Metsch in a slightly more particular setting (cf. [4, p. 284]), but was never published. Let B^{∗} be a blocking set of W(2n − 1, p) which does not contain a blocking set of strictly smaller size. Then there is a generator of W(2n − 1, p) that meets B^{∗} in a unique point x. Each of the α := p^{n−1} + ⋯ + p^{2} + p other points of this generator sees in its quotient a blocking set of a W(2n − 3, p), so besides x at least β := p^{n−3}(s(W(3, p)) − 1) further points. As every point of B^{∗} \ {x} is counted at most γ := p^{n−2} + ⋯ + p + 1 times, we have |B^{∗} \ x| ≥ αβ/γ.

This proves the main result. □

Note that the geometrical results of this section are still valid if we replace the field 𝔽_{p} by 𝔽_{q} when q is any odd prime power.

The minimal size of a blocking set of W(3, 3), respectively W(3, 5), equals 11, respectively 29 — see [1, Remark 10]. So for these cases, we have 10 · 3^{n−2} + 1 ≤ s(W(2n − 1, 3)) ≤ 11 · 3^{n−2} and 28 · 5^{n−2} + 1 ≤ s(W(2n − 1, 5)) ≤ 29 · 5^{n−2}.

References

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[2] A. Klein and K. Metsch. New results on covers and partial spreads of polar spaces, Innov. Incidence Geom. 1 (2005), 19–34.

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[9] J. A. Thas. Ovoids, spreads and m-systems of finite classical polar spaces, in: Surveys in Combinatorics, 2001 (Sussex), London Math. Soc. Lecture Note Series 288, Cambridge University Press, Cambridge (2001), 241–267.

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[11] E. Yalçin. Private communication, December 2006/January 2007.