Subgroup rigidity in finite-dimensional group algebras over $p$-groups

SUBGROUP FAMILIES CONTROLLING p-LOCAL FINITE GROUPS

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode “conjugacy” relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we exa...

Group Algebras of Finite Groups as Lie Algebras

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group G, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of G. MSC 2000 : 20C15,17B99.

Finite Dimensional Pointed Hopf Algebras over S

Finite groups with a unique subgroup of order p

This is a toy example to illustrate some of the techniques of fusion and transfer as described in Gorenstein’s text Finite Groups. Finite groups with a unique subgroup of order p (p an odd prime) are classified in terms of a cyclic p′-extension of relatively simple combinatorial data. This note is intended to address a slightly misstated exercise in somewhat more detail and to be a toy example ...

On $m^{th}$-autocommutator subgroup of finite abelian groups

Let $G$ be a group and $Aut(G)$ be the group of automorphisms of‎ ‎$G$‎. ‎For any natural‎ number $m$‎, ‎the $m^{th}$-autocommutator subgroup of $G$ is defined‎ ‎as‎: ‎$$K_{m} (G)=langle[g,alpha_{1},ldots,alpha_{m}] |gin G‎,‎alpha_{1},ldots,alpha_{m}in Aut(G)rangle.$$‎ ‎In this paper‎, ‎we obtain the $m^{th}$-autocommutator subgroup of‎ ‎all finite abelian groups‎.