Weakly closed implies relatively normal: Difference between revisions

Latest revision as of 20:35, 11 February 2009

This article gives the statement and possibly, proof, of an implication relation between two subgroup-of-subgroup properties. That is, it states that every subgroup-of-subgroup satisfying the first subgroup-of-subgroup property (i.e., weakly closed subgroup) must also satisfy the second subgroup-of-subgroup property (i.e., relatively normal subgroup)
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Statement

Suppose $H \leq K \leq G$ are such that $H$ is a Weakly closed subgroup (?) of $K$ relative to $G$ . Then, $H$ is a Normal subgroup (?) of $K$ . In other words, $H$ is a Relatively normal subgroup (?) in $K$ with respect to $G$ .

@@ Line 1: / Line 1: @@
+{{subofsubgroup property implication|
+stronger = weakly closed subgroup|
+weaker = relatively normal subgroup}}
 ==Statement==
-Suppose <math>H \le K \le G</math> are such that <math>H</math> is a [[fact about::weakly closed subgroup]] of <math>K</math> relative to <math>G</math>. Then, <math>H</math> is a [[fact about::normal subgroup]] of <matH>K</math>.
+Suppose <math>H \le K \le G</math> are such that <math>H</math> is a [[fact about::weakly closed subgroup]] of <math>K</math> relative to <math>G</math>. Then, <math>H</math> is a [[fact about::normal subgroup]] of <math>K</math>. In other words, <math>H</math> is a [[fact about::relatively normal subgroup]] in <math>K</math> with respect to <math>G</math>.