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

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:

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