Large subgroups in finite groups

TL;DR

This paper investigates conditions under which large subgroups, especially nilpotent and abelian ones, can be systematically identified within finite groups, enhancing understanding of their internal subgroup structure.

Contribution

It provides general theorems for producing large subgroups in finite groups and explores the existence of large nilpotent and abelian subgroups in soluble groups.

Findings

Established methods for constructing large subgroups in finite groups.

Identified conditions for large nilpotent subgroups in soluble groups.

Analyzed properties of large abelian subgroups in finite groups.

Abstract

Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) \leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing large subgroups in finite groups (see Theorems A and C). We also consider the more specialised problems of finding large (non-abelian) nilpotent as well as abelian subgroups in soluble groups.