Prove that every subgroup of an abelian group is a normal subgroup.

Prove that every subgroup of an abelian group is a normal subgroup.

The definition of a normal group is:

A group $H\leq G$ is a normal subgroup if for any $g\in G$, the set $gH$ equals the set $Hg$. Equivalently, you can also demand $H=gHg^{-1}$.

Now, take a subgroup $H$ of an abelian group $G$. Take any element $x\in gHg^{-1}$. By definition, this element must equal $ghg^{-1}$ for some $h\in H$.

Now you have $x=ghg^{-1}$, where $g\in G$ and $h\in H$. Can you prove $x\in H$?

By definition, a normal subgroup $N\triangleleft G$ is one that $Ng=gN\ \forall g\in G$; that is, $\forall g\in G,n\in N\ \exists\,n'\in N: ng=gn'$ and $\forall g\in G,n\in N\ \exists\,n'\in N: n'g=gn$. For each of those, choose $g,n$ and let $n'=n$; because $G$ is abelian, you have the equation desired.