Research ArticleNo Access

Finite groups with weakly $ℋ C$ -embedded subgroups

M. Ramadan

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

https://doi.org/10.1142/S0219498820500930Cited by:3 (Source: Crossref)

Abstract

Let $G$ be a finite group and $H$ a subgroup of $G$ . We say that $H$ is an $ℋ$ -subgroup of $G$ if $N_{G} (H) \cap H^{g} \leq H$ for all $g \in G$ . We say that $H$ is weakly $ℋ 𝒞$ -embedded in $G$ if $G$ has a normal subgroup $T$ such that $H^{G} = H T$ and $N_{T} (H) \cap H^{g} \leq H$ for all $g \in G,$ where $H^{G}$ is the normal closure of $H$ in $G$ . For each prime $p$ dividing the order of $G,$ let $P$ be a Sylow $p$ -subgroup of $G$ . We fix a subgroup of $P$ of order $d$ with $1 < d < | P |$ and study the structure of $G$ under the assumption that every subgroup of $P$ of order $p^{n} d$ $(n = 0, 1)$ is weakly $ℋ 𝒞$ -embedded in $G$ . Our results improve and generalize several recent results in the literature.

Communicated by M. L. Lewis

Keywords:

AMSC: 20D10, 20D15, 20D20

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