Critical subgroup

This article is about a subgroup property related to the Classification of finite simple groups

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with critical group

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

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

Facts

1. Every group of prime power order has a critical subgroup. Further information: Thompson's critical subgroup theorem, Analysis of Thompson's critical subgroup theorem
2. A group 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.
3. A critical subgroup that is also extraspecial as a group must equal the whole group. For full proof, refer: extraspecial and critical implies whole group

Relation with other properties

Stronger properties

• Abelian critical subgroup
• Constructibly critical subgroup

Weaker properties

• Self-centralizing characteristic subgroup
• Self-centralizing normal subgroup
• Commutator-in-center subgroup
• Class two normal subgroup
• Coprime automorphism-faithful subgroup: For full proof, refer: Critical implies coprime automorphism-faithful

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. Further information: Thompson's critical subgroup theorem, Analysis of Thompson's critical subgroup theorem
2. 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.
3. A critical subgroup that is also extraspecial as a group must equal the whole group. For full proof, refer: extraspecial and critical implies whole group
4. Abelian Frattini subgroup implies centralizer is critical

Metaproperties

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.