Fully transitive $p$-groups with finite first Ulm subgroup

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...

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 ...

Finite Groups with NR−Subgroup

Let G be a finite group. Yakov Berkovic investigated the following concept: A subgroup H of G is called NR−subgroup with respect to G if A = AG ⋂ H for any subgroup A H.In particulary,called a finite group G NN−group if its any subgroup is either normal subgroup or NR−subgroup of G. In fact, all groups with order p -p3 are NN -group,where p is a prime. In this paper, the nature and structure of...

Finite Groups with p-Sylow Coverings

Finite Transitive Permutation Groups and Finite Vertex-Transitive Graphs

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups. In this chapter we explore some of the ways the two theories have influenced each other. On the one hand each finite transitive permutation group corresponds to several vertex-transitive graphs, namely the generalised orbital graphs which we shall discuss below. On the other hand, ...