Critical subgroup

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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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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:

  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. \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).

Relation with other properties

Stronger properties

Weaker properties

Group properties satisfied

Any critical subgroup satisfies the following group properties:

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. 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
  3. 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.
  4. 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
  5. 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.