Some notes on σ-soluble groups and σ-subnormality

Abstract

Let G be a finite group and $\sigma =\left\{{\sigma }_i\right|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\underset{i\in I}{\cup }{\sigma }_i$ and ${\sigma }_i\cap {\sigma }_j=\varnothing$ for all $i\ne j$. The natural numbers n and m are called σ-coprime if $\sigma (n)\cap \sigma (m)=\varnothing$. The group G is said to be: σ-primary if G is a σ_i-group for some $i\in I$; σ-soluble if either G = 1 or every chief factor of G is σ-primary. A subgroup H of G is called σ-subnormal in G if there is a subgroup chain $H=H_0\le H_1\le \cdots \le H_t=G$ such that either $H_{i-1}$ is normal in H_i or $H_i/{(H_{i-1})}_{H_i}$ is σ-primary for all $i=1,\dots ,t$. In this paper, we show that G is σ-soluble provided G satisfies the following conditions: (1) $G=A_1{A}_2=A_1{A}_3=A_2{A}_3$, where A₁, A₂, A₃ are all σ-soluble; (2) the three indices $|G:N_G(A_1^{{\mathfrak{N}}_{\sigma }})|,|G:N_G(A_2^{{\mathfrak{N}}_{\sigma }})|,|G:N_G(A_3^{{\mathfrak{N}}_{\sigma }})|$ are pairwise σ-coprime.

And we prove that if G is σ-soluble and A is a subgroup of G, then A is σ-subnormal in G if and only if $|AB{|}_{{\sigma }_i}$ divides $|G{|}_{{\sigma }_i}$ for every Hall σ_i-subgroup B of G and all ${\sigma }_i\in \sigma (G)$. We also state and prove a σ-nilpotency criterion for G and a characterization of the σ-Fitting subgroup of G, which are related to this observation.

Keywords:

• Finite group
• Hall σ-subgroups
• σ-nilpotent groups
• σ-soluble groups
• σ-subnormal subgroups

2020 Mathematics Subject Classification:

• 20D10
• 20D15
• 20D20

Additional information

Funding

Research was supported by the NSFC of China (No. 12001526, 12201252) and Natural Science Foundation of Jiangsu Province, China (No. BK20200626, BK20210442).