On Fischer subgroups of finite groups
DOI:
https://doi.org/10.37863/umzh.v73i7.6192Keywords:
Fitting set, Fischer set, F-injector, Fischer F-subgroup of G.Abstract
UDC 512.542
Let F be a Fitting set of a group G, π be a nonempty set of primes, and L≤G.
In this case, F is called a Fischer π-set of G
if conditions L∈F, K⊴ 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.
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