The determination of abelian Hall subgroups by a conjugacy class structure.

Wolfgang Kimmerle; Robert Sandling

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Let $G$ be any group and let $A$ be an abelian quasinormal subgroup of $G$ . If $n$ is any positive integer, either odd or divisible by $4$ , then we prove that the subgroup $A^{n}$ is also quasinormal in $G$ .

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We describe the finite groups satisfying one of the following conditions: all maximal subgroups permute with all subnormal subgroups, (2) all maximal subgroups and all Sylow $p$ -subgroups for $p < 7$ permute with all subnormal subgroups.