On the Cohomology of Central Frattini Extensions

We use topological methods to compute the mod p cohomology of certain p-groups. More precisely we look at central Frattini extensions of elementary abelian by elementary abelian groups such that their defining k-invariants span the entire image of the Bockstein. We show that if p is sufficiently large, then the mod p cohomology of the extension can be explicitly computed as an algebra

Lifting Lie algebras over the residue field of a discrete valuation ring

Studies among other things, the question of whether a Lie algebra over Z/(p^k)Z lifts to one over Z/(p^(k+1))Z. An obstruction theory is developed and examples of Fp-Lie algebras which don't lift to Lie algebras over Z/p^2Z are discussed. An example of an application of the result: A Fp-Lie algebra L with H^3(L, ad)=0 will lift to a p-adic Lie algebra

Quadratic Maps and Bockstein Closed Group Extensions

We study central extensions E of elementary abelian 2-groups by elementary abelian 2-groups. Associated to such an extension is a quadratic map which determines the extension uniquely. The components of the map determine a quadratic ideal in a polynomial algebra and we say that the extension is Bockstein closed if this ideal is invariant under the Bockstein operator. We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this we show for example that quadratic maps induced from the fundamental quadratic map on gl_n, Q(A)=A+A^2 yield Bockstein closed extensions. We also show that this condition is equivalent to a certain liftability of the extension and under certain conditions, to the fact that Q corresponds to a 2-power map of a restricted 2-Lie algebra. For the class of 2-groups coming from these type of extensions, under a 2-power exact condition we also compute the mod 2 cohomology ring of the corresponding groups - this includes the upper triangular congruence subgroups among other examples.Comment: To appear, Transactions of the A.M.

On commuting and non-commuting complexes

In this paper we study various simplicial complexes associated to the commutative structure of a finite group G. We define NC(G) (resp. C(G)) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets in G. We observe that NC(G) has only one positive dimensional connected component, which we call BNC(G), and we prove that BNC(G) is simply connected. Our main result is a simplicial decomposition formula for BNC(G) which follows from a result of A. Bjorner, M. Wachs and V. Welker on inflated simplicial complexes. As a corollary, we obtain that if G has a nontrivial center or if G has odd order, then the homology group H_{n-1}(BNC(G)) is nontrivial for every n such that G has a maximal noncommuting set of order n

Bockstein Closed 2-Group Extensions and Cohomology of Quadratic Maps

A central extension of the form $E: 0 \to V \to G \to W \to 0$, where $V$ and $W$ are elementary abelian 2-groups, is called Bockstein closed if the components q_i \in H^*(W, \FF_2) of the extension class of $E$ generate an ideal which is closed under the Bockstein operator. In this paper, we study the cohomology ring of $G$ when $E$ is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of $G$ has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg-Moore spectral sequence. We also find an interpretation of the second page of the Bockstein spectral sequence in terms of a new cohomology theory that we define for Bockstein closed quadratic maps $Q : W \to V$ associated to the extensions $E$ of the above form.Comment: 31 pages. To appear in Journal of Algebr