Critical subgroup: Difference between revisions
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* <math>\Phi(H) \le Z(H)</math>, viz the [[Frattini subgroup]] is contained inside the [[center]] | * <math>\Phi(H) \le Z(H)</math>, viz the [[Frattini subgroup]] is contained inside the [[center]] | ||
* <math>[G,H] \le Z(H)</math> | * <math>[G,H] \le Z(H)</math> | ||
* <math>C_G(H)= Z(H)</math> | * <math>C_G(H)= Z(H)</math> (i.e., <math>H</math> is a [[self-centralizing subgroup]] of <math>G</math>) | ||
==Facts== | ==Facts== | ||
Every [[group of prime power order]] has a critical subgroup. {{further|[[Thompson's critical subgroup theorem]]}} | Every [[group of prime power order]] has a critical subgroup. {{further|[[Thompson's critical subgroup theorem]]}} | ||
Revision as of 21:49, 6 July 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
Definition with symbols
Let be a group of prime power order.
A subgroup of is said to be critical if it is characteristic in , and the following three conditions hold:
- , viz the Frattini subgroup is contained inside the center
- (i.e., is a self-centralizing subgroup of )
Facts
Every group of prime power order has a critical subgroup. Further information: Thompson's critical subgroup theorem