Central factor is centralizer-closed

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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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Statement

Verbal statement

The Centralizer (?) of a central factor of the whole group is also a central factor of the whole group.

Proof

Given: H \le G is a subgroup such that HC_G(H) = G.

To prove: C_G(H) is a central factor of G: C_G(H)C_G(C_G(H)) = G.

Proof: Clearly, H \le C_G(C_G(H)), so C_G(H)H \le C_G(H)C_G(C_G(H)). Since HC_G(H) = G, we have C_G(H)H = G (for more, see permuting subgroups), so G \le C_G(H)C_G(C_G(H)), forcing G = C_G(H).