profinite group
A profinite group is what you get by stitching together a whole tower of finite groups into one compact object. Imagine knowing a group only through its finite quotients, finer and finer, and assembling all that approximate knowledge into a single limit. The result is a topological group that is compact and totally disconnected, and it is the only honest way to make sense of infinite Galois groups, where the natural topology cannot be ignored.
Formally, a profinite group is an inverse (projective) limit of an inverse system of finite groups, each given the discrete topology; equivalently, a compact Hausdorff totally disconnected topological group. Its open subgroups are exactly the finite-index ones that are open, and they form a neighborhood basis of the identity. Every profinite group is the inverse limit of its finite quotients G/N over open normal subgroups N, recovering it from finite data.
The motivating example is the absolute Galois group Gal(K-sep/K) = lim Gal(L/K) over finite Galois subextensions L; the Krull topology makes the fundamental theorem of Galois theory work for infinite extensions, matching closed subgroups with intermediate fields. Cohomology of profinite groups must be continuous: cochains are required to be locally constant, and H^n is computed as the colimit of the cohomology of finite quotients. This continuous cohomology is precisely what Galois cohomology uses.
The Galois group of the finite-field closure, Gal(F̄_p / F_p), is the profinite completion Ẑ = lim Z/nZ — generated topologically by the Frobenius x |-> x^p. Likewise the p-adic integers Z_p = lim Z/p^k Z are a profinite group under addition.
The absolute Galois group of a finite field is Ẑ.
Not every subgroup of finite index in a profinite group is open in general, but for topologically finitely generated profinite groups a theorem of Nikolov–Segal guarantees that every finite-index subgroup is open, so the algebraic and topological structures coincide.