New criteria for \(\sigma \)-subnormality in \(\sigma \)-solvable finite groups

Let \(\mathbb {P}\) $\mathbb{P}$ be the set of all prime numbers, I be a set and \(\sigma = \lbrace \sigma _i \mid i \in I \rbrace \) $\sigma =\{{\sigma }_i\mid i\in I\}$ be a partition of \(\mathbb {P}\) $\mathbb{P}$ . A finite group is said to be \(\sigma \) $\sigma$ -primary if it is a \(\sigma _i\) ${\sigma }_i$ -group for some \(i \in I\) $i\in I$ , and we say that a finite group is \(\sigma \) $\sigma$ -solvable if all its chief factors are \(\sigma \) $\sigma$ -primary. A subgroup H of a finite group G is said to be \(\sigma \) $\sigma$ -subnormal in G if there is a chain \(H = H_0 \le H_1 \le \dots \le H_n = G\) $H=H_0\le H_1\le \cdots \le H_n=G$ of subgroups of G such that \(H_{i-1}\) $H_{i-1}$ is normal in \(H_i\) $H_i$ or \(H_i/(H_{i-1})_{H_i}\) $H_i/{(H_{i-1})}_{H_i}$ is \(\sigma \) $\sigma$ -primary for all \(1 \le i \le n\) $1\le i\le n$ . Given subgroups H and A of a \(\sigma \) $\sigma$ -solvable finite group G, we prove two criteria for H to be \(\sigma \) $\sigma$ -subnormal in \(\langle H, A \rangle \) $⟨H,A⟩$ . Our criteria extend classical subnormality criteria of Fumagalli [5], which themselves generalize a classical subnormality criterion of Wielandt [13].