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

Definition with symbols

A subgroup $H$ of a group $G$ is said to be critical if the following three conditions hold:

• $\mathrm{\Phi }(H)\le Z(H)$, viz the Frattini subgroup is contained inside the center
• $[G,H]\le Z(H)$
• $C_G(H)\le Z(H)$

Facts

Every group of prime power order has a critical subgroup.