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.
VIEW: Definitions built on this | Facts about this: (facts closely related to Critical subgroup, all facts related to Critical subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions

Definition

Definition with symbols

Let $G$ be a group of prime power order.

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

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

Facts

Every group of prime power order has a critical subgroup. Further information: Thompson's critical subgroup theorem, Analysis of Thompson's critical subgroup theorem