# Critical subgroup

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*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

## 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:

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

### Definition with symbols

Let be a group of prime power order (or more generally, a possibly infinite p-group).

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

- , i.e., the Frattini subgroup is contained inside the center (i.e., is a Frattini-in-center group).
- (i.e., is a commutator-in-center subgroup of ).
- (i.e., is a self-centralizing subgroup of ).

## Relation with other properties

### Stronger properties

### Weaker properties

- Weakly critical subgroup: The same conditions, except that we drop the condition of characteristicity in the whole group.
- 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:

## 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` - More generally, every infinite p-group that is also abelian-by-nilpotent has a critical subgroup.
`Further information: Analogue of critical subgroup theorem for infinite abelian-by-nilpotent p-group` - 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.
- 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` - 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.