Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients. Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space. Equivalently, one can define a profinite group to be a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. In categorical terms, this is a special case of a (co)filtered limit construction. The group of p-adic integers Zp under addition is profinite (in fact procyclic). It is the inverse limit of the finite groups Z/pnZ where n ranges over all natural numbers and the natural maps Z/pnZ -> Z/pmZ (n>=m) are used for the limit process. The topology on this profinite group is the same as the topology arising from the p-adic valuation on Zp.

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