Constructibly critical subgroup

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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 $G/K$

Relation with other properties

Stronger properties

• Abelian critical subgroup

Weaker properties

• Critical subgroup

Group properties satisfied

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

Metaproperties

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.