I am not exactly a group theorist, so this may be well-known.


Let $G$ be a finitely generated group such that the cardinality of minimal generating sets of $G$ is bounded above.
Does it follow that $G$ is finite?


This is true if $G$ is abelian, but I have no idea about the general case.



Minimal generating sets of groups
Just to expand my comment, a [Tarski Monster](http://en.wikipedia.org/wiki/Tarski_monster_group) is an infinite group in which, for some fixed prime $p$, all proper nontrivial subgroups have order $p$. It was proved by Olshanskii in 1979 that they exist for all primes $p>10^{75}$.


A set of $p+1$ distinct elements of such a group cannot all lie in the same subgroup of order $p$, so they must generate the whole group, and so we have an upper bound of $p+1$ on the size of minimal generating sets.


It would interesting to know whether there are any less exotic examples.


