Cocentrality is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., cocentral subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Statement with symbols

If $H$ is a cocentral subgroup of $K$, and $K$ is a cocentral subgroup of $G$, then $H$ is a cocentral subgroup of $G$.

Definitions used

Cocentral subgroup

Further information: Cocentral subgroup

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

Related facts

• Cocentral implies centralizer-dense

Proof

Given: A group $G$, a cocentral subgroup $K$ of $G$, a cocentral subgroup $H$ of $K$. In other words, $HZ(K)=K$, and $KZ(G)=G$.

To prove: $H$ is cocentral in $G$, i.e., $HZ(G)=G$.

Proof: Consider $C_{G}(Z(K))$. Clearly, $Z(G)\leq C_{G}(Z(K))$, and $K\leq C_{G}(Z(K))$, because $Z(K)$ centralizes $K$. Thus, $KZ(G)\leq C_{G}(Z(K))$, so $C_{G}(Z(K))=G$. Thus, $Z(K)\leq Z(G)$. Thus, $HZ(K)\leq HZ(G)$, so $K\leq HZ(G)$, and thus $G=KZ(G)\leq HZ(G)$. Thus, <mathG = HZ(G)</math>.