Constructibly critical subgroup

Definition
Let $$G$$ be a group of prime power order.

A subgroup $$H$$ of $$G$$ is termed a constructibly critical subgroup if it satisfies the following two conditions:


 * 1) The center of $$H$$, say $$K := Z(H)$$, is maximal among Abelian characteristic subgroups
 * 2) $$H$$ is the intersection of $$C_G(K)$$ and the inverse image in $$G$$ of the subgroup $$\Omega_1(Z(G/K))$$ of $$G/K$$

Stronger properties

 * Weaker than::Abelian critical subgroup

Weaker properties

 * Stronger than::Critical subgroup

Group properties satisfied

 * Frattini-in-center group
 * Group in which every Abelian characteristic subgroup is central

Left realization
A group of prime power order can be realized as a constructibly critical subgroup if it satisfies the following two conditions:


 * It is a Frattini-in-center group: Its Frattini subgroup is contained in its center.
 * It is a group in which every Abelian characteristic subgroup is central.

Further, any such group is the unique constructibly critical subgroup of itself.