Critical subgroup: Difference between revisions

Revision as of 19:47, 17 November 2008

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

Symbol-free definition

A subgroup of a group of prime power order 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 $G$ 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:

$Φ (H) \leq Z (H)$ , i.e., the Frattini subgroup is contained inside the center (i.e., $H$ is a Frattini-in-center group).
$[G, H] \leq Z (H)$ (i.e., $H$ is a commutator-in-center subgroup of $G$ ).
$C_{G} (H) = Z (H)$ (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
A group 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

Relation with other properties

Stronger properties

Weaker properties

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
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

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.

@@ Line 6: / Line 6: @@
 ==Definition==
+===Symbol-free definition===
+A [[subgroup]] of a [[group of prime power order]] is termed a '''critical subgroup''' if it is [[characteristic subgroup|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 of two subgroups|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 <math>G</math> be a [[group of prime power order]].
-A subgroup <math>H</math> of <math>G</math> is said to be '''critical''' if it is [[characteristic subgroup|characteristic]] in <math>G</math>, and the following three conditions hold:
+A subgroup <math>H</math> of <math>G</math> is said to be '''critical''' if it is [[defining ingredient::characteristic subgroup|characteristic]] in <math>G</math>, and the following three conditions hold:
-# <math>\Phi(H) \le Z(H)</math>, viz the [[Frattini subgroup]] is contained inside the [[center]] (i.e., <math>H</math> is a [[defining ingredient::Frattini-in-center group]]).
+# <math>\Phi(H) \le Z(H)</math>, i.e., the [[Frattini subgroup]] is contained inside the [[center]] (i.e., <math>H</math> is a [[defining ingredient::Frattini-in-center group]]).
 # <math>[G,H] \le Z(H)</math> (i.e., <math>H</math> is a [[defining ingredient::commutator-in-center subgroup]] of <math>G</math>).
 # <math>C_G(H)= Z(H)</math> (i.e., <math>H</math> is a [[self-centralizing subgroup]] of <math>G</math>).