On the partial $\mathcal{L}$-$\Pi$-property of subgroups of finite groups

Abstract

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the partial $\mathcal{L}$-$\Pi$-property in $G$ if $H⊴G$, or if $|G/K:{\mathrm{N}}_{G/K}(HK/K)|$ is a $\pi (HK/K)$-number for any $G$-chief factor of type $H^G/K$ with $H_G\le K$. In this paper, we investigate the structure of finite groups under the assumption that some subgroups of prime power order satisfy the partial $\mathcal{L}$-$\Pi$-property.

arXiv admin note: substantial text overlap with arXiv:2310.09814, arXiv:2404.01254, arXiv:2304.11451; text overlap with arXiv:1301.6361 by other authors