On Fischer subgroups of finite groups - Mao , Yu.; Ma, X.; Vorob'ev, N. Т.; Karaulova, T. B. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 512.542Let $\mathscr{F}$ be a Fitting set of a group $G,$ $π$ be a nonempty set of primes, and $L≤ G.$In this case, $\mathscr{F}$ is called a Fischer $π$-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 $p\in π$ 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 $L≤ F.$It is said that a Fitting set $\mathscr{F}$ of $G$ is $π$-saturated if $\mathscr{F} = \{H ≤ G : H/H_\mathscr{F} \in \mathfrak{E}_{π'} \},$ where $\mathfrak{E}_{π'}$ is the class of all $π'$-groups.In this paper, under the condition that $\mathscr{F}$ is a $π$-saturated Fischer $π$-set of a $π$-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 $π'$-subgroup of $G.$