A profinite group is a totally disconnected compact topological group. Jarden and Lubotzky had shown in 2008 that if two finitely generated profinite groups are elementarily equivalent in the language of groups, then they are in fact already isomorphic. Around the same time, Frohn had studied the theory of abelian profinite groups in the Cherlin-van den Dries-Macintyre language of inverse systems and reached a similar conclusion for so-called small abelian profinite groups. A common generalization of these two results was given recently by Helbig. I will explain these results and discuss some related questions concerning elementary equivalence (in the language of groups and in the language of inverse systems) and isomorphism (as abstract groups and as profinite groups).