Extension of Carter subgroups in $\pi$-separable groups

TL;DR

This paper proves that in $ ext{pi}$-separable groups, there exists a conjugacy class of subgroups that generalize Carter subgroups, which are self-normalizing nilpotent subgroups, extending known results from soluble groups.

Contribution

It extends the concept of Carter subgroups to $ ext{pi}$-separable groups, showing the existence of a conjugacy class of such subgroups beyond soluble groups.

Findings

Existence of a conjugacy class of Carter-like subgroups in $ ext{pi}$-separable groups.

Generalization of Carter subgroup properties to broader class of groups.

Connection between Carter subgroups and nilpotent projectors in this context.

Abstract

Let $\pi$ be a set of primes. We show that $\pi$-separable groups have a conjugacy class of subgroups which specialize to Carter subgroups, i.e. self-normalizing nilpotent subgroups, or equivalently, nilpotent projectors, when specializing to soluble groups.