Every Subgroup of an Abelian Group is Normal
Theorem
Let \(G\) be an abelian group and \(H\) a subgroup. Then \(H \trianglelefteq G\), that is, \(H\) is normal in \(G\).
Proof
Let \(g \in G\) and \(h \in H\). Now, because we are in a commutative group, we have
\[\begin{align*}
g \ast h \ast g^{-1} &= g \ast g^{-1} \ast h \\
&= \mathrm{id} \ast h \\
&= h \in H \\
\end{align*}\]
and thus \(H\) is normal.