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A topological group G is profinite if it is compact and totally disconnected.Equivalently, G is the inverse limit of a surjective system of finite groupscarrying the discrete topology. We discuss how to represent a countably basedprofinite group as a point in a Polish space. Then we study the complexity ofisomorphism using the theory of Borel reducibility in descriptive set theory.For topologically finitely generated profinite groups this complexity is thesame as the one of identity for reals. In general, it is the same as thecomplexity of isomorphism for countable graphs.

The complexity of isomorphism between countably based profinite groups