On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

Finite alperin 2-groups with cyclic second commutants

An Alperin group is a group in which every 2-generated subgroup has a cyclic commutant. Previously, we constructed examples of finite Alperin 2-groups with second commutant isomorphic to Z ₂ or Z ₄. Here, it is proved that for any natural n, there exists a finite Alperin 2-group whose second commutant is isomorphic to Z _2n .     

On maximal abelian subgroups in finite 2-groups

We determine here up to isomorphism the structure of any finite nonabelian 2-group G in which every two distinct maximal abelian subgroups have cyclic intersection. We obtain five infinite classes of such 2-groups (Theorem 1.1). This solves for p = 2 the problem Nr. 521 stated by Berkovich (in preparation). The more general problem Nr. 258 stated by Berkovich (in preparation) about the structure of finite nonabelian p-groups G such that A ∩ B = Z(G) for every two distinct maximal abelian subgroups A and B is treated in Theorems 3.1 and 3.2. In Corollary 3.3 we get a new result for an arbitrary finite 2-group. As an application of Theorems 3.1 and 3.2, we solve for p = 2 a problem of Heineken-Mann (Problem Nr. 169 stated in Berkovich, in preparation), classifying finite 2-groups G such that A/Z(G) is cyclic for each maximal abelian subgroup A (Theorem 4.1).      

A characterization of elementary abelian 2-groups

In this note we give a characterization of elementary abelian 2-groups in terms of their maximal sum-free subsets.     

On 2-groups with finite subgroups of rank 2

A classification is achieved of the 2-groups all of whose finite subgroups are generated by two elements.     

Cohomology with coefficients in stacks of abelian 2-groups

Cohomology of a topological space with coefficients in stacks of abelian 2-groups is invented. This theory extends the classical sheaf cohomology. An application is given to twisted sheaves.     

Minimal non-quaternion-free finite 2-groups

We classify minimal non-quaternion-free finite 2-groups which possess a nonmodular proper subgroup.     

Representations of elementary abelian p-groups and finite subgroups of fields

Suppose $\mathbb{F}$ is a field of prime characteristic p and E is a finite subgroup of the additive group $(\mathbb{F},+)$. Then E is an elementary abelian p-group. We consider two such subgroups, say E and $E^{\prime }$, to be equivalent if there is an $\alpha \in {\mathbb{F}}^{\times }:=\mathbb{F}?\{0\}$ such that $E=\alpha E^{\prime }$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.