Finite groups with σ-abnormal or σ-subnormal σ-primary subgroups

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 $i\ne j.$ A set $\mathcal{H}$ of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of $\mathcal{H}$ is a Hall σ_i-subgroup of G for some i and $\mathcal{H}$ contains exactly one Hall σ_i-subgroup of G for every ${\sigma }_i\in \sigma (G).$ A group is said to be σ-primary if it is a finite σ_i-group for some i. A subgroup A of G is said to be: σ-subnormal in G if there is a subgroup chain $A=A_0\le A_1\le \cdots \le A_t=G$ such that either $A_{i-1}$ is normal in A_i or $A_i/{(A_{i-1})}_{A_i}$ is σ-primary for all $i=1,2,\dots ,t;$ σ-abnormal in G if $K/L_K$ is not σ-primary whenever $A\le L<K\le G$ and L is a maximal subgroup of K. In this article, we study the structure of a finite group in which σ-primary cyclic subgroups are σ-abnormal or σ-subnormal in G. We also describe the structure of a finite group G which has a complete Hall σ-set $\{H_1,H_2,\dots ,H_t\}$ such that for any $A_i\le H_i,$ G has a subgroup A of order $\left|A_1\right|\left|A_2\right|\cdots \left|A_t\right|$ and A is σ-abnormal or σ-subnormal in G.

Keywords:

• σ-Abnormal subgroup
• finite group
• σ-primary subgroup
• σ-subnormal subgroup

2020 Mathematics Subject Classification:

• 20D10
• 20D15
• 20D20

Additional information

Funding

Research supported by the Fundamental Research Funds for the Central Universities (JUSRP121048).