Normal implies permutable

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property must also satisfy the second subgroup property
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Statement

Verbal statement

Any normal subgroup of a group is a permutable subgroup.

Symbolic statement

Let $G$ be a group and $H$ a normal subgroup of $G$. Then $H$ is a permutable, or quasinormal, subgroup of $G$. In other words $HK=KH$ for any subgroup $K$ of $G$.

Property-theoretic statement

The subgroup property of being normal is stronger than the subgroup property of being permutable.

Definitions used

Normal subgroup

Further information: Normal subgroup A subgroup $H$ of a group $G$ is a normal subgroup if for any $g\in G$, $gH=Hg$ (viz, the left cosets are the same as the right cosets).

Permutable subgroup

Further information: Permutable subgroup A subgroup $H$ of a group $G$ is a permutable subgroup if for any subgroup $K\le G$, $HK=KH$.

Proof

Let $H$ be a normal subgroup of $G$. We need to show that $H$ is permutable in $G$.

Let $K$ be any subgroup of $G$. For every $g\in K$, $Hg=gH$. Now we have:

$HK={\bigcup }_{g\in K}Hg$

and

$KH={\bigcup }_{g\in K}gH$

Since $Hg=gH\mathrm{\forall }g\in K$, we conclude that $HK=KH$.

Notice that the above proof does not anywhere use the fact that $K$ is a subgroup.