Centrally large subgroup

Definition
Let $$P$$ be a group of prime power order. A subgroup $$A$$ of $$P$$ is termed a centrally large subgroup or CL-subgroup if it satisfies the following equivalent conditions:


 * $$|A||Z(A)| \ge |B||Z(B)|$$ for any subgroup $$B$$ of $$P$$.
 * 1) $$C_P(A) \le A$$ (i.e., $$A$$ is a defining ingredient::self-centralizing subgroup of $$P$$) and $$A$$ is a defining ingredient::centralizer-large subgroup of $$P$$.

Weaker properties

 * Stronger than::Self-centralizing subgroup
 * Stronger than::Centralizer-large subgroup