A contribution to the theory of $σ$-properties of a finite group

• DOI: 10.48550/arxiv.2404.00004
• Publication Date: 2024-01-01
• Authors: Liu, A-Ming; Guo, Wenbin; Safonov, Vasily G.; Skiba, Alexander N.
• Keywords: 20D10, 20D15, 20D30; FOS: Mathematics; Group Theory (math.GR)

19 pages<br/>We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every maximal subgroup $V$ of the generalized Fitting subgroup $F^{*}(G)$ of $G$; a finite soluble group $G$ is a $PST$-group (that is, Sylow permutability is a transitive relation on $G$) if and only if $G$ has a normal subgroup $D$ such that $G/D$ is nilpotent and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every subnormal subgroup $A$ of $G$.<br/>