A Generalization of $\sigma $-Permutability

Abstract

Throughout this paper, all groups are finite and G always denotes a finite group; $\sigma $ is some partition of the set of all primes $\mathbb {P}$. A group G is said to be $\sigma $-primary if G is a $\pi $-group for some $\pi \in \sigma $. A $\pi $-semiprojector of G [29] is a subgroup H of G such that HN/N is a maximal $\pi $-subgroup of G/N for all normal subgroups N of G. Let $\Pi \subseteq \sigma $. Then we say that $ {\mathcal {X}} = \{X_ {1}, \ldots , X_ {t} \} $ is a $ \Pi $-covering subgroup system for a subgroup H in G if all members of the set $ {\mathcal {X}} $ are $ \sigma $-primary subgroups of G and for each $ \pi \in \Pi $ with $\pi \cap \pi (H)\ne \emptyset $ there are an index i and a $ \pi $-semiprojector U of H such that $ U \le X_ {i}$. We study the embedding properties of subgroups H of G under the hypothesis that G has a $ \Pi $-covering subgroup system $ {\mathcal {X}}$ such that H permutes with $X^{x}$ for all $X\in {\mathcal {X}}$ and $x\in G$. Some well-known results are generalized.

Keywords

Finite group / $\sigma $-nilpotent group')">$\sigma $-nilpotent group / $\Pi $-semiprojector')">$\Pi $-semiprojector / $\Pi $-covering subgroup system')">$\Pi $-covering subgroup system / $\sigma $-subnormal subgroup')">$\sigma $-subnormal subgroup

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Funding

NNSF of China(12171126)

NNSF of China(12171126)