New criteria for $$\sigma $$ -subnormality in $$\sigma $$ -solvable finite groups

Let \(\mathbb {P}\) be 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 finite group is said to be \(\sigma \)-primary if it is a \(\sigma _i\)-group for some \(i \in I\), and we say that a finite group is \(\sigma \)-solvable if all its chief factors are \(\sigma \)-primary. 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 \(H_{i-1}\) is normal in \(H_i\) or \(H_i/(H_{i-1})_{H_i}\) is \(\sigma \)-primary for all \(1 \le i \le n\). Given subgroups H and A of a \(\sigma \)-solvable finite group G, we prove two criteria for H to be \(\sigma \)-subnormal in \(\langle H, A \rangle \). Our criteria extend classical subnormality criteria of Fumagalli [5], which themselves generalize a classical subnormality criterion of Wielandt [13].