Taras Banakh today told me about a solution of a group theory problem by means of topological algebra. If I remembered it right, it is the following.
For a cardinal $\kappa$ as $S(\kappa)$ we denote the group of all bijections of $\kappa$. Prove that if $\kappa<\lambda$ then a group $S(\lambda)$ is not embeddable into a group $S(\kappa)$. If $| S(\lambda)|=2^\lambda>2^\kappa=| S(\kappa)|$ then the non-embeddability follows from the size comparison, which implies the proof under GCH. But in order to obtain a proof in ZFC, Taras Banakh introduced a cardinal invariant $H(G)$ of a group $G$, which equals to a minimal weight of a Hausdorff group topology on the group $G$. Clearly, if $G’$ is a subgroup of the group $G$ then $H(G’)\le H(G)$. So the results follows from the inequality $H(S(\kappa))< H(S(\lambda))$.