Finite groups with some σ-primary subgroups sσ-quasinormal

Abstract

Let G be a finite group, $\sigma =\left\{{\sigma }_i\right|i\in I\}$ a partition of the set of all primes $\mathbb{P}$ and $\sigma (G)=\left\{{\sigma }_i\right|{\sigma }_i\cap \pi (|G|)\ne \varnothing \}.$ 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 $i\in I$ and $\mathcal{H}$ contains exactly one Hall σ_i-subgroup of G for every ${\sigma }_i\in \sigma (G).$ G is said to be σ-full if G possesses a complete Hall σ-set. We say a subgroup H of G is sσ-quasinormal (supplement-σ-quasinormal) in G if there exists a σ-full subgroup T of G such that G = HT and H permutes with every Hall σ_i-subgroup of T for all ${\sigma }_i\in \sigma (T).$ In this article, we obtain some results about the sσ-quasinormal subgroups and use them to determine the structure of finite groups. In particular, some new criteria of p-nilpotency, solubility, supersolubility of a group are obtained.

Keywords:

• Finite group
• maximal subgroup
• soluble group
• sσ-quasinormal subgroup
• supersoluble group

MATHEMATICS SUBJECT CLASSIFICATION:

• 20D10
• 20D15
• 20D20
• 20D35

Additional information

Funding

Research was supported by the NNSF of China (11771409) and Anhui Initiative in Quantum Information Technologies (AHY150200).