Critical subgroup

Symbol-free definition
A subgroup of a group of prime power order (or more generally, of a possibly infinite p-group) is termed a critical subgroup if it is characteristic in the whole group and satisfies the following three conditions:


 * 1) The subgroup is a Frattini-in-center group: Its Frattini subgroup is contained in its center.
 * 2) The subgroup is a commutator-in-center subgroup: Its commutator with the whole group is contained in its center.
 * 3) The subgroup is a self-centralizing subgroup: Its centralizer in the whole group is contained in it.

Definition with symbols
Let $$G$$ be a group of prime power order (or more generally, a possibly infinite p-group).

A subgroup $$H$$ of $$G$$ is said to be critical if it is characteristic in $$G$$, and the following three conditions hold:


 * 1) $$\Phi(H) \le Z(H)$$, i.e., the Frattini subgroup is contained inside the center (i.e., $$H$$ is a defining ingredient::Frattini-in-center group).
 * 2) $$[G,H] \le Z(H)$$ (i.e., $$H$$ is a defining ingredient::commutator-in-center subgroup of $$G$$).
 * 3) $$C_G(H)= Z(H)$$ (i.e., $$H$$ is a self-centralizing subgroup of $$G$$).

Stronger properties

 * Weaker than::Abelian critical subgroup
 * Weaker than::Constructibly critical subgroup
 * Weaker than::c-closed critical subgroup

Weaker properties

 * Stronger than::Weakly critical subgroup: The same conditions, except that we drop the condition of characteristicity in the whole group.
 * Stronger than::Self-centralizing characteristic subgroup
 * Stronger than::Self-centralizing normal subgroup
 * Stronger than::Commutator-in-center subgroup
 * Stronger than::Class two normal subgroup
 * Stronger than::Coprime automorphism-faithful subgroup:

Group properties satisfied
Any critical subgroup satisfies the following group properties:


 * Frattini-in-center group
 * Group of nilpotence class two

Facts

 * 1) Every group of prime power order has a critical subgroup.
 * 2) More generally, every infinite p-group that is also abelian-by-nilpotent has a critical subgroup.
 * 3) A group of prime power order can arise as a critical subgroup of some group, if and only if it is a Frattini-in-center group. Further, any Frattini-in-center group is a critical subgroup of itself.
 * 4) A critical subgroup that is also extraspecial as a group must equal the whole group.
 * 5) Abelian Frattini subgroup implies centralizer is critical

Left realization
A group of prime power order can arise as a critical subgroup of some group, if and only if it is a Frattini-in-center group (in other words, its Frattini subgroup is contained in its center). Further, any Frattini-in-center group is a critical subgroup of itself.

Right realization
By Thompson's critical subgroup theorem, every group of prime power order possesses a critical subgroup.