The generalised Fitting subgroup of a profinite group

TL;DR

This paper extends the concept of the generalized Fitting subgroup from finite groups to a class of profinite groups, exploring structural implications and properties in this broader context.

Contribution

It introduces a new class of profinite groups generalizing the generalized Fitting subgroup and investigates their structural properties.

Findings

The generalized Fitting subgroup always contains its own centralizer in this class.

Profinite groups in this class are structured as abelian extensions of automorphism groups.

The paper establishes foundational properties of these generalized subgroups in profinite groups.

Abstract

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of profinite groups which generalises this phenomenon, and explore some consequences for the structure of profinite groups.