gr.group theory - Does a group have a unique pro-p topology?

Your question, in the finitely generated case, is sort of answered above.

In the infinitely generated case, however, this is a research question I have been looking at for the last year or so. I'm in the middle of redrafting a paper dealing with the abelian case.

If $G$ is any finitely-generated pro-$p$ group, by Serre, any pro-$p$ topology must have all finite index subgroups open, so the answer here is yes. This is not too hard to see. A result of Nikolov and Segal (2006) showed the same holds for profinite groups.

There are two different notions of unique topology you could mean here. If $f:Gto H$ is an abstract isomorphism between two pro-p groups, we have three possibilities:

(1) $f$ must be continuous.
(2) $f$ is not continuous, but $G,H$ are isomorphic as pro-$p$ groups.
(3) $G,H$ are not isomorphic as pro-$p$ groups.

Which case this is is entirely dependant on the algebraic structure of $G$.

The first case is equivalent to saying all automorphisms of $G$ are continuous. As above, any finitely-generated pro-$p$, indeed, profinite groups have this property. However not every group here must be finitely generated. By consideration of centralisers, which, as kernels of word maps, must be closed in any profinite topology, one can see that any (unrestricted Cartesian) product of finite centreless groups has this property. Moreover, considering centralisers will also give this result for Branch Groups.

If we assume the Generalised Continuum Hypothesis fails, we get uninteresting examples of non-isomorphic abstractly ismorphic pro-$p$ groups - the (unrestricted Cartesian) product of $aleph$, $beth$ copies of $C_p$ with $2^aleph=2^beth$, for example. It is more interesting to look for examples which do not depend on this. Hence it makes sense to look at countably-based profinite groups - those which are the inverse limit of a countable collection of finite groups, first.

Tyler Lawson gives a good example above of this, but it is not too hard to see that these groups are still abstractly isomorphic.

More interestingly, if $G$ is a countably-based abelian pro-$p$ group with infinite exponent torsion group, then $G$ is abstractly isomorphic to a product of cyclic $p$-groups, and there are uncountably many different pro-$p$ topologies on $G$, which give rise to uncountably many isomorphism classes of pro-$p$ groups.

My paper showing this is mathematically complete but being redrafted. A version should appear on the ArXiV in the next few days. In the meantime, I can e-mail a copy on request.