Cocentrality is transitive

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 $$G$$.

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, $$Z(G) \le C_G(Z(K))$$, and $$K \le C_G(Z(K))$$, because $$Z(K)$$ centralizes $$K$$. Thus, $$KZ(G) \le C_G(Z(K))$$, so $$C_G(Z(K)) = G$$. Thus, $$Z(K) \le Z(G)$$. Thus, $$HZ(K) \le HZ(G)$$, so $$K \le HZ(G)$$, and thus $$G = KZ(G) \le HZ(G)$$. Thus, <mathG = HZ(G).