# Profinite group

A topological group that is the projective limit of an inverse system of finite discrete groups $G _ {i}$, $i \in I$( where $I$ is a pre-ordered directed set). The profinite group $G$ is denoted by $\lim\limits _ \leftarrow G _ {i}$. As a subspace of the direct product $\prod _ {i \in I } G _ {i}$, endowed with the compact topology (a neighbourhood base of the identity is given by the system of kernels of the projections $\prod _ {i \in I } G _ {i} \rightarrow G _ {j}$), it is closed and hence compact.

### Examples.

1) Let $I$ be the set of integers larger than zero with the natural order relation, and let $G _ {i} = \mathbf Z / p ^ {i} \mathbf Z$. Suppose that $\tau _ {i} ^ {i+1} : G _ {i+1} \rightarrow G _ {i}$ is the natural epimorphism, and put

$$\tau _ {i} ^ {j} = \tau _ {i} ^ {i+1} \tau _ {i+1} ^ {i+2} \dots \tau _ {j-1} ^ {j}$$

for all $i < j$. Then $\lim\limits _ \leftarrow G _ {i}$ is the (additive) group of the ring $\mathbf Z _ {p}$ of $p$- adic integers.

2) Every compact analytic group over a $p$- adic number field (e.g. $\mathop{\rm SL} _ {n} ( \mathbf Z _ {p} )$) is profinite as a topological group.

3) Let $G$ be an abstract group and let $\{ {H _ {i} } : {i \in I } \}$ be the family of its normal subgroups of finite index. On $I$ one introduces the relation $\leq$, putting $i \leq j$ if $H _ {i} \supseteq H _ {j}$. This relation turns $I$ into a pre-ordered directed set. Associate to $i \in I$ the group $G / H _ {i}$, and to each pair $( i , j )$, $i \leq j$, the natural homomorphism $\tau _ {i} ^ {j} : G / H _ {j} \rightarrow G / H _ {i}$. One obtains the profinite group $\widehat{G} = \lim\limits _ \leftarrow G / H _ {i}$, called the profinite group completion of $G$. It is the separable completion of $G$( cf. Separable completion of a ring) for the topology defined by the subgroups of finite index. The kernel of the natural homomorphism $G \rightarrow \widehat{G}$ is the intersection of all subgroups of finite index. In this construction one can consider, instead of the family of all normal subgroups of finite index, only those whose index is a fixed power of a prime number $p$. The corresponding group is denoted by $\widehat{G} _ {p}$, and is a pro- $p$- group.

4) Profinite groups naturally arise in Galois theory of (not necessarily finite) algebraic extensions of fields in the following way. Let $K / k$ be a Galois extension and suppose that $\{ {K _ {i} / k } : {i \in I } \}$ is the family of all finite Galois extensions of $k$ lying in $K$. Then $K = \cup _ {i \in I } K _ {i}$, and one can introduce on $I$ the relation $\leq$ by putting $i \leq j$ if $K _ {i} \subseteq K _ {j}$. The set $I$ then becomes pre-ordered. Let $\mathop{\rm Gal} ( K _ {i} / k )$ be the Galois group of $K _ {i} / k$. To every pair $( i , j ) \in I \times I$, $i \leq j$, one naturally associates the homomorphism

$$\tau _ {i} ^ {j} : \mathop{\rm Gal} K _ {j} / k \rightarrow \mathop{\rm Gal} K _ {i} / k .$$

The corresponding profinite group $\lim\limits _ \leftarrow \mathop{\rm Gal} ( K _ {i} / k )$ is isomorphic to $\mathop{\rm Gal} ( K / k )$, thus $\mathop{\rm Gal} ( K / k )$ can be considered as a profinite group. The system $\{ \mathop{\rm Gal} ( K _ {i} / k ) \} _ {i}$ forms in $\mathop{\rm Gal} ( K / k )$ a neighbourhood base of the identity (cf. Galois topological group). This construction has a generalization in algebraic geometry in the definition of the fundamental group of a scheme.

A profinite group can be characterized as a compact totally-disconnected group (cf. Compact group), as well as a compact group that has a system of open normal subgroups forming a neighbourhood base of the identity. The cohomology theory of profinite groups (cf. Cohomology of groups; Galois cohomology) plays an important role in modern Galois theory.

How to Cite This Entry:
Profinite group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Profinite_group&oldid=51493
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article