Critical subgroup: Difference between revisions

Latest revision as of 00:05, 13 February 2009

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:

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 (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:

$Φ (H) \leq Z (H)$ , 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 $G$ ).

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.

@@ Line 1: / Line 1: @@
+{{stdnonbasicdef}}
 {{cfsg-subgroup property}}
+{{nottobeconfusedwith|[[critical group]]}}
+==Definition==
-{{nottobeconfusedwith|[[critical group]]}}
+===Symbol-free definition===
-{{stdnonbasicdef}}
+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 subgroup|characteristic]] in the whole group and satisfies the following three conditions:
-==Definition==
+# 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===
-A subgroup <math>H</math> of a group <math>G</math> is said to be '''critical''' if the following three conditions hold:
+Let <math>G</math> be a [[group of prime power order]] (or more generally, a possibly infinite [[p-group]]).
-* <math>\Phi(H) \le Z(H)</math>, viz the [[Frattini subgroup]] is contained inside the [[center]]
+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>[G,H] \le Z(H)</math>
-* <math>C_G(H) \le Z(H)</math>
+# <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>).
+==Relation with other properties==
+===Stronger properties===
+* [[Weaker than::Abelian critical subgroup]]
+* [[Weaker than::Constructibly critical subgroup]]
+* [[Weaker than::c-closed critical subgroup]]
+===Weaker properties===
+* [[Stronger than::Weakly critical subgroup]]: The same conditions, except that we drop the condition of characteristicity in the whole group.
+* [[Stronger than::Self-centralizing characteristic subgroup]]
+* [[Stronger than::Self-centralizing normal subgroup]]
+* [[Stronger than::Commutator-in-center subgroup]]
+* [[Stronger than::Class two normal subgroup]]
+* [[Stronger than::Coprime automorphism-faithful subgroup]]: {{proofat|[[Critical implies coprime automorphism-faithful]]}}
+===Group properties satisfied===
+Any critical subgroup satisfies the following group properties:
+* [[Frattini-in-center group]]
+* [[Group of nilpotence class two]]
 ==Facts==
-Every [[group of prime power order]] has a critical subgroup.
+# Every [[group of prime power order]] has a critical subgroup. {{further|[[Thompson's critical subgroup theorem]], [[Analysis of Thompson's critical subgroup theorem]]}}
+# More generally, every infinite [[p-group]] that is also [[abelian-by-nilpotent group|abelian-by-nilpotent]] has a critical subgroup. {{further|[[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 group|extraspecial]] as a group must equal the whole group. {{proofat|[[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.