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 Hj 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.
