Inverse Limits and Profinite Groups

Standard examples of inverse limits arise from sequences of groups, with maps between them: for instance, if we have the sequence Gn = Z/pZ for n ≥ 0, with the natural quotient maps πn+1 : Gn+1 → Gn, the inverse limit consists of tuples (g0, g1, . . . ) ∈ ∏ n≥0Gn such that πn+1(gn+1) = gn for all n ≥ 0. This is a description of the p-adic integers Zp. It is clear that more generally if the Gn are any groups and πn+1 : Gn+1 → Gn any homomorphisms, we can define a notion of the inverse limit group in the same way. However, we will make a more general definition.