# Constructibly critical subgroup

From Groupprops

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## Definition

Let be a group of prime power order.

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

- The center of , say , is maximal among Abelian characteristic subgroups
- is the intersection of and the inverse image in of the subgroup of

## Relation with other properties

### Stronger properties

### Weaker properties

### Group properties satisfied

## 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.