Profinite group

In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.

The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists ${\displaystyle d\in \mathbb {N} }$ such that every group in the system can be generated by ${\displaystyle d}$ elements.^[1] Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems.^[2]

To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profinite group.

Important examples of profinite groups are the additive groups of ${\displaystyle p}$-adic integers and the Galois groups of infinite-degree field extensions.

Every profinite group is compact and totally disconnected. A non-compact generalization of the concept is that of locally profinite groups. Even more general are the totally disconnected groups.