On σ-subnormal subgroups and products of finite groups

Abstract

Suppose that $ \sigma = \{ {\sigma}_{i} : i \in I \} $ is a partition of the set $ \mathbb{P} $ of all primes. A subgroup $ A $ of a finite group $ G $ is said to be $ \sigma $-subnormal in $ G $ if $ A $ can be joined to $ G $ by a chain of subgroups $ A = A_{0} \subseteq A_{1} \subseteq \cdots \subseteq A_{n} = G $ such that either $ A_{i-1} $ normal in $ A_{i} $ or $ A_{i}/ Core_{A_{i}}(A_{i-1}) $ is a $ {\sigma}_{j} $-group for some $ j \in I $, for every $ 1\leq i \leq n $. A $ \sigma $-subnormality criterion related to products of subgroups of finite $ \sigma $-soluble groups is proved in the paper. As a consequence, a characterisation of the $ \sigma $-Fitting subgroup of a finite $ \sigma $-soluble group naturally emerges.

Keywords

• finite group
• σ-soluble group
• σ-subnormal subgroup
• products of groups