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

Abstract

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].