On Fischer subgroups of finite groups

Abstract

UDC 512.542

Let $\mathscr{F}$ be a Fitting set of a group $G,$ $\pi$ be a nonempty set of primes, and $L\leq G.$
In this case, $\mathscr{F}$ is called a Fischer $\pi$-set of $G$
if conditions $L\in\mathscr{F},$ $K\unlhd L,$ and $H/K$ is a $p$-subgroup of $L/K$ for a prime $p\in \pi$ imply necessarily that $H \in \mathscr{F}.$
It is said that a subgroup $F$ of $G$ is a Fischer $\mathscr{F}$-subgroup of $G$
if the following conditions hold:
1) $F \in \mathscr{F};$
2) if $L$ is an $\mathscr{F}$-subgroup of $G$ normalized by $F,$ then $L\leq F.$
It is said that a Fitting set $\mathscr{F}$ of $G$ is $\pi$\emph{-saturated} if $\mathscr{F} = \{H \leq G : H/H_\mathscr{F} \in \mathfrak{E}_{\pi'} \},$ where $\mathfrak{E}_{\pi'}$ is the class of all $\pi'$-groups.

In this paper, under the condition that $\mathscr{F}$ is a $\pi$-saturated Fischer $\pi$-set of a $\pi$-soluble group $G,$
we prove that a subgroup $V$ of $G$ is an $\mathscr{F}$-injector of $G$ if and only if $V$ is a Fischer $\mathscr{F}$-subgroup of $G$ containing a Hall $\pi'$-subgroup of $G.$