Weakly closed subgroup: Difference between revisions

Revision as of 20:45, 2 March 2009

This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.
View other such properties

Definition

Suppose $H\leq K\leq G$ . Then, $H$ is termed weakly closed in $K$ relative to $G$ if, for any $g\in G$ such that $gHg^{-1}\leq K$ , we have $gHg^{-1}\leq H$ .

There is a related notion of weakly closed subgroup for a fusion system.

Relation with other properties

Stronger properties

Strongly closed subgroup

Weaker properties

Relatively normal subgroup: For full proof, refer: Weakly closed implies normal in middle subgroup
Conjugation-invariantly relatively normal subgroup when the big group is a finite group: For full proof, refer: Weakly closed implies conjugation-invariantly relatively normal in finite group

Facts

Weakly closed implies normal in middle subgroup: If $H\leq K\leq G$ and $H$ is weakly closed in $K$ relative to $G$ , then $H$ is a normal subgroup of $K$ .
Weakly normal implies weakly closed in intermediate nilpotent: If $H\leq K\leq G$ , with $H$ a weakly normal subgroup of $G$ , and $K$ a nilpotent group, then $H$ is a weakly closed subgroup of $K$ .

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	* [[Weakly closed implies normal in middle subgroup]]: If <math>H \le K \le G</math> and <math>H</math> is weakly closed in <math>K</math> relative to <math>G</math>, then <math>H</math> is a [[normal subgroup]] of <math>K</math>.		* [[Weakly closed implies normal in middle subgroup]]: If <math>H \le K \le G</math> and <math>H</math> is weakly closed in <math>K</math> relative to <math>G</math>, then <math>H</math> is a [[normal subgroup]] of <math>K</math>.
	* [[Weakly normal implies weakly closed in intermediate nilpotent]]: If <math>H \le K \le G</math>, with <math>H</math> a [[~~paranormal~~ subgroup]] of <math>G</math>, and <math>K</math> a [[nilpotent group]], then <math>H</math> is a weakly closed subgroup of <math>K</math>.		* [[Weakly normal implies weakly closed in intermediate nilpotent]]: If <math>H \le K \le G</math>, with <math>H</math> a [[weakly normal subgroup]] of <math>G</math>, and <math>K</math> a [[nilpotent group]], then <math>H</math> is a weakly closed subgroup of <math>K</math>.