On Wielandt's zipper lemma and \(\sigma\)-subnormal subgroups of finite groups

Abstract

Let \(\mathbb{P}\) denote the set of all prime numbers, I be a set and \(\sigma = \lbrace \sigma_i \mid i \in I \rbrace\) be a partition of \(\mathbb{P}\). A subgroup H of a finite group G is said to be \(\sigma\)-subnormal in G if there is a chain \(H = H_0 \le H_1 \le \dots \le H_n = G\) of subgroups of G such that, for each \(1 \le j \le n\), the subgroup \(H_{j-1}\) is normal in H_j or \(H_j/(H_{j-1})_{H_j}\) is a \(\sigma_i\)-group for some \(i \in I\). If \(\sigma\) is the partition of \(\mathbb{P}\) into subsets of size one, then the concept of \(\sigma\)-subnormality reduces to the familiar concept of subnormality. In recent years, many results about subnormal subgroups have been extended to results about \(\sigma\)-subnormal subgroups. This line of research is continued in the present note by proving a \(\sigma\)-version of Wielandt's zipper lemma.