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.



Non-commutator in simple group?
For infinite groups, you may find examples here:


<http://arxiv.org/abs/arXiv:0909.2294>


In fact, in the reference above you may find examples of infinite simple groups with infinite commutator width. In other words, examples of simple groups $G$ such that for every $n\in\mathbb{N}$ there exists an element $g\in G$ that is not the product of less than $n$ commutators.


