Cocentral implies central factor

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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., cocentral subgroup) must also satisfy the second subgroup property (i.e., central factor)
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Statement

Any cocentral subgroup of a group is a central factor.

Definitions used

Cocentral subgroup

Further information:

A subgroup H of a group G is a cocentral subgroup if HZ(G) = G where Z(G) is the center of G.

Central factor

Further information: central factor

A subgroup H of a group G is a central factor if HC_G(H) = G where C_G(H) is the centralizer of H in G.

Related facts

Proof

Given: A group G, a subgroup H of G such that HZ(G) = G where Z(G) is the center of G.

To prove: HC_G(H) = G where C_G(H) is the centralizer of H in G.

Proof: We have Z(G) \le C_G(H), because any element of the center centralizes every element of H. Thus, HZ(G) \le HC_G(H). Since G = HZ(G), we obtain G = HC_G(H).