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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WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with critical group
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
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
- 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:
- 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
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.
By Thompson's critical subgroup theorem, every group of prime power order possesses a critical subgroup.