On σ-subnormal closure

Abstract

Let $\sigma =\{{\sigma }_i:i\in I\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup A of a finite group G is called σ-subnormal in G if there is a chain of subgroups $A=A_0\subseteq A_1\subseteq \cdots \subseteq A_n=G$ with $A_{i-1}$ normal in A_i or $A_i/{\text{Core}}_{A_i}(A_{i-1})$ is a ${\sigma }_j$-group for some $j\in I,$ for every $1\le i\le n.$ In this paper, the description of the unique smallest σ-subnormal subgroup of a σ-soluble group containing a given subgroup is obtained.

Keywords:

• Finite group
• σ-soluble group
• σ-subnormal subgroup

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 20D10
• 20D20

Additional information

Funding

This Project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. G: 1493-130-1440. The authors, therefore, acknowledge with thanks DSR for technical and financial support.