Hi,

For a group $G$, we say that $x\in G$ is a commutator if there exists $a,b \in G$ such that $x=a^{1}b^{-1}ab$, and we say that $x$ is a **non-commutator** if there is no $a,b \in G$ such that $x=a^{1}b^{-1}ab$.

Does there exist a non abelian simple group $G$ (finite or not) such that $G$ has at least one non-commutator?

I tried with $-I_n$ in $\mathrm{PSL}(n,q)$ but no luck.

Thanks.