I recently stumbled over the example in
<http://ysharifi.wordpress.com/2010/03/09/a-uniquely-divisible-non-abelian-group/>
of a non-abelian group $G$ with the property that for all natural numbers $n$ and elements $x\in G$ there is $y\in G$ such that $x=y^n$. (In this particular example $y$ is unique but I don't care about that.) I will call such groups divisible, although I'm not sure if this term is used for abelian groups exclusively.


I wonder wether there are examples of non-abelian divisible groups, that satisfy additional assumptions. I am in particular interested in non-abelian divisible


$\bullet$ simple groups,


$\bullet$ finitely generated groups,


$\bullet$ finitely presented groups,


$\bullet$ groups satisfying all or some of the above properties at the same time.


Unfortunately I don't manage to find examples, but I'd appreciate any help.



Non-abelian divisible groups
See:


V. S. Guba, Finitely generated complete groups, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986),
883-924. 


for an interesting 2-generated example. (Furthermore, roots in Guba's examples are unique.) Note that a group $G$ is called *complete* if for any non-trivial word $u(x\_1,\cdots,x\_m)$ and every $g\in G$, the equation $u(x\_1,\cdots,x\_m)=g$ has a solution. In your case, the words $u$ are of the form $x^n$. I do not know if Guba's group is simple, but it does have a (nontrivial) simple quotient, which is necessarily verbal, and, in particular, divisible. 


None of these groups is finitely-presented and, I think, the problem of existence of fp divisible groups is open. The philosophical reason why is the following. Call an infinite finitely-generated group "exotic" if it satisfies some bizarre, seemingly impossible property, e.g., being a torsion group, containing very few conjugacy classes, being divisible, etc. The most common method for constructing *exotic* groups $G$ is by direct limit of a sequence of (relatively) hyperbolic groups $G\_k$ which are quotients of a single group $G\_0$. If $G$ were finitely presented, it would be isomorphic to one of the groups $G\_k$ and, hence, non-exotic. 


Mark Sapir will probably have more comments on this. 


