True or false: If $N$ is a normal subgroup of $G$, then $gng^{-1 }= n$ for all $n \in N$ and $g \in G$. Justify

Question

True or false: If $N \triangleleft G$ , then $gng^{-1 }= n$ for all $n \in N$ and $g \in G$. Justify your answer

Isn't this the definition of a normal subgroup? Is it asking me to prove the definition?

Answer 1

Note that a subgroup $H$ is normal if for all $g \in G$, $gHg^{-1}=H$, i.e. the subgroup $H$ as a whole is fixed under conjugation. This does not mean every element of $H$ is fixed. Try to find a counterexample yourself by looking at the examples of normal subgroups from your notes or your course textbook. You only need to find a single element in the subgroup not fixed by conjugation.

Hint. In abelian groups, $ghg^{-1}=gg^{-1}h=1h=h$ for all $g \in G$, so do not try looking in abelian groups!