The number of compact Hausdorff group topologies on a given group strongly depends on the algebraic structure of the group.

For example, any finite-dimensional torus $(\mathbb R/\mathbb Z)^n$ has $2^{\mathfrak c}$ automorphisms, among which only finitely many continuous. This implies that $(\mathbb R/\mathbb Z)^n$ has $2^{\mathfrak c}$ pairwise incomparable compact Hausdorff group topologies.

On the other hand, for any odd $n$ the group $SO(n,\mathbb R)$ has a unique compact Hausdorff topology, according to a classical result (1933) of van der Waerden.

A compact topological group $G$ is called *van der Waerden* (or else *self-bohrifying*) if every group homomorphism $G\to K$ to a compact Hausdorff topological group is continuous. It is easy to see that van der Waerden groups admit a unique compact Hausdorff topology. This paper of Hart and Kunen contains many examples of van der Waerden (=self-bohrifying) compact Hausdorff topological groups.

In particular, by Lemma 5.20 in the mentioned paper of Hart and Kunen, a countable product $\prod_{k\in\omega}G_k$ of finite groups is self-bohrifying if

1) No group occurs infinitely often in the list $(G_n)_{n\in\omega}$;

2) Each $G_k$ is either $A_{k_n}$ or $PSL(j_k, q_k)$ or $SL(j_k, q_k)$, where $\sup_k j_k=\infty$.

group topologieson the given group? $\endgroup$ – Yemon Choi Feb 10 at 3:18