Perfect subnormal implies subnormal-permutable

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., perfect subnormal subgroup) must also satisfy the second subgroup property (i.e., subnormal-permutable subgroup)
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Any perfect subnormal subgroup of a group is subnormal-permutable: it permutes with any subnormal subgroup of the whole group. Thus, any perfect subnormal subgroup is a subnormal-permutable subnormal subgroup.

Facts used

  1. Orthogonal subnormal subgroups permute


The proof follows from fact (1), and the observation that a perfect group is orthogonal to any group.