Non-abelian divisible groups 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.
 A: 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.  
A: One often reads that divisible groups are important since they help us understanding the structure of abelian groups, for they are all and the only injectives in the usual category of abelian groups (which is, of course, undeniable). Yet, I find that the non-abelian case is, if possible, even more interesting. A 'natural' example of a non-commutative divisible group is the group of units of Hamilton's quaternions; afak, the result is due to I. Niven [1]. On another hand, it was recently proved on this forum that the general linear group of degree $n$ over an algebraically closed field $\mathbb K$ is divisible iff $\mathbb K$ has zero characteristic (see here), and I've just posed a similar question for ${\rm SL}_n(\mathbb K)$ (see here)
[1]  I. Niven, The Roots of a Quaternion, The Amer. Math. Monthly, Vol. 49, No. 6 (Jun. - Jul., 1942), pp. 386-388.
A: Any connected, compact Lie group has this property. To get convinced about this, take an element $g \in G$ where $G$ is a compact Lie group, then $g$ can be put inside a torus $T$ in $G$, and a torus being a direct product of finitely many copies of circle, $S^1$, the result now follows.
Caution: One needs to prove the result that every element of $G$ has an $n$-th root to show that every $g \in G$ can be put in a torus. The proof uses degree argument of a smooth map between manifolds. 
