Suppose $G$ is a compact $T_2$ group. Can there be other compact $T_2$ topologies on $G$ which also turn $G$ into a topological group? ($T_2$ refers to the Hausdorff separation axiom)
Take the circle group $G=S^1=\mathbb R/\mathbb Z$. Any non-continuous automorphism of $\mathbb R$ which fixes pointwise the subgroup $\mathbb Z$ passes to the quotient and gives an automorphism $f$ of the abstract group $G$, which is not continuous. Now define a topology on $G$ so that a set $U$ is open iff $f(U)$ is open in the usual topology. This new topology is of course Hausdorff and compact, but it is different to the usual topology.