Central factor is centralizer-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property)
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Given: is a subgroup such that .
To prove: is a central factor of : .
Proof: Clearly, , so . Since , we have (for more, see permuting subgroups), so , forcing .