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.

References

[1]	J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303
[2]	H. Koch, "Galoissche Theorie der $p$-Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)
[3]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986) MR0911121 Zbl 0645.12001 Zbl 0153.07403