Injectors in $\pi$-separable groups

TL;DR

This paper proves that in $ ext{pi}$-separable groups, there exists a conjugacy class of $ ext{F}$-injectors within certain Fitting classes, generalizing known results for soluble groups.

Contribution

It establishes the existence and conjugacy of $ ext{F}$-injectors in $ ext{pi}$-separable groups, extending classical results beyond soluble groups.

Findings

Existence of conjugacy class of $ ext{F}$-injectors in $ ext{pi}$-separable groups

Generalization of injectors from soluble to $ ext{pi}$-separable groups

Identification of suitable Fitting classes for $ ext{pi}$-separable groups

Abstract

Let $\pi$ be a set of primes. We show that $\pi$-separable groups have a conjugacy class of $\mathfrak F$-injectors for suitable Fitting classes $\mathfrak F$, which coincide with the usual ones when specializing to soluble groups.