Let $G$ be a group, $G'=[G, G]$.



> 
> "Note that it is not necessarily true that the commutator subgroup 
>  $G'$ of $G$ consists entirely of
>  commutators $[x, y], x, y \in G$ (see [107] for some finite group examples)." 
> 
> 
> 


Quoted from <http://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf> page 8.


Anybody can provide the examples? I can't find the book [107].



Commutator subgroup does not consist only of commutators?
The problem is whether the commutator subgroup may contain elements that are not commutators. 
One example are the free groups. For instance, in the free group of rank $4$, freely generated by $x$, $y$, $z$, and $w$, the element $[x,y][z,w]$ of the commutator subgroup cannot be written in the form $[a,b]$ for some $a,b$ in the group.


The smallest finite examples are groups of order 96; there's two of them, nonisomorphic to each other. (This was a result in Robert Guralnick's thesis). See [this Math Stack exchange question](https://math.stackexchange.com/questions/7811/derived-subgroups-and-commutators) for a description of these groups, and some references.


