Show that G', the commutator subgroup of G, is normal in G. Prove that any subgroup A with G' $\subseteq$ A $\subseteq$ G is normal in G.

Question

Show that G', the commutator subgroup of G, is normal in G.
Prove that any subgroup A with G' $\subseteq$ A $\subseteq$ G is normal in G.

So the definition of the commutator subgroup is that;

In the group G, let G' be the subgroup generated by the set {$xyx^{-1}y^{-1}$ : x,y in G}.

So to show it is normal in G I must show that ($xyx^{-1}y^{-1})^{-1}G'xyx^{-1}y^{-1}$=G' but I just can't manipulate it to work.

Thanks

Answer 1

A subgroup $N$ of $G$ is normal if $gng^{-1}$ is in $N$ for all $g\in G$ and $n\in N$. You can use this to check normality of the derived subgroup.

Answer 2

Suppose $g \in A$. You know that $k= hgh^{-1}g^{-1} \in A$, because $k= hgh^{-1}g^{-1} \in G' \subset A$. Hence $hgh^{-1} = kg \in A$.

Note that this proof is shorter than the standard proof that $G'$ is normal in $G$.