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 $w(G)$ of a group $G$, which equals to a minimal weight of a Hausdorff group topology on the group $G$. Clearly, if $H$ is a subgroup of the group $G$ then $w(H)\le w(G)$. Banakh and Mildenberger proved that $w(G)=\kappa$ for any group $G\subset S(\kappa)$ containing the subgroup $Alt(\kappa)$ consisting of finitely supported even permutations of an infinite cardinal $\kappa$. So, for any infinite cardinals $\kappa<\lambda$ the inequality $w(S(\kappa))=\kappa<\lambda=w(Alt(\kappa))$ implies that $Alt(\lambda)$ is not isomorphic to a subgroup of $S(\kappa)$.
