Weakly closed subgroup: Difference between revisions

Latest revision as of 21:23, 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

Normalizer-relatively normal subgroup: For full proof, refer: Weakly closed implies normalizer-relatively normal
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 $K$ .

@@ Line 3: / Line 3: @@
 ==Definition==
-Suppose <math>H \le K \le G</math>. Then, <math>H</math> is termed '''weakly closed''' in <math>K</math> relative to <math>G</math> if, for any <math>g \in G</math> such that <math>gHg^{-1} \le K</math>, we have <math>gHg^{-1} = H</math>.
+Suppose <math>H \le K \le G</math>. Then, <math>H</math> is termed '''weakly closed''' in <math>K</math> relative to <math>G</math> if, for any <math>g \in G</math> such that <math>gHg^{-1} \le K</math>, we have <math>gHg^{-1} \le H</math>.
 There is a related notion of [[weakly closed subgroup for a fusion system]].
@@ Line 12: / Line 12: @@
 * [[Weaker than::Strongly closed subgroup]]
+===Weaker properties===
+* [[Stronger than::Normalizer-relatively normal subgroup]]: {{proofat|[[Weakly closed implies normalizer-relatively normal]]}}
+* [[Stronger than::Relatively normal subgroup]]: {{proofat|[[Weakly closed implies normal in middle subgroup]]}}
+* [[Stronger than::Conjugation-invariantly relatively normal subgroup]] when the big group is a [[finite group]]: {{proofat|[[Weakly closed implies conjugation-invariantly relatively normal in finite group]]}}
 ==Facts==
 * [[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>.
-* [[Paranormal 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>.