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A generalisation of Schenkman's theorem | | Stefanos Aivazidis
; Ina N. Safonova
; Alexander N. Skiba
; | | Date: |
17 Sep 2020 | | Abstract: | Let $G$ be a finite group and let $mathfrak{F}$ be a hereditary saturated
formation. We denote by $mathbf{Z}_{mathfrak{F}}(G)$ the product of all
normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$
is $mathfrak{F}$-central in $G$, that is, [ (H/K)
times
(G/mathbf{C}_{G}(H/K)) in mathfrak{F}. ]A subgroup $A leq G$ is said to be
$mathfrak{F}$-subnormal in the sense of Kegel, or $K$-$mathfrak{F}$-subnormal
in $G$, if there is a subgroup chain [ A = A_0 leq A_1 leq ldots leq A_n =
G ] such that either $A_{i-1} rianglelefteq A_{i}$ or $A_i / (A_{i-1})_{A_i}
in mathfrak{F}$ for all $i = 1, ldots , n$. In this paper, we prove the
following generalisation of Schenkman’s Theorem on the centraliser of the
nilpotent residual of a subnormal subgroup: Let $mathfrak{F}$ be a hereditary
saturated formation and let $S$ be a $K$-$mathfrak{F}$-subnormal subgroup of
$G$. If $mathbf{Z}_{mathfrak{F}}(E) = 1$ for every subgroup $E$ of $G$ such
that $S leq E$ then $mathbf{C}_{G}(D) leq D$, where $D = S^{mathfrak{F}}$
is the $mathfrak{F}$-residual of $S$. | | Source: | arXiv, 2009.08145 | | Services: | Forum | Review | PDF | Favorites | |
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