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A topological group that is the [[Projective limit|projective limit]] of an inverse system of finite discrete groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750502.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750503.png" /> is a pre-ordered directed set). The profinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750504.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750505.png" />. As a subspace of the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750506.png" />, endowed with the compact topology (a neighbourhood base of the identity is given by the system of kernels of the projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750507.png" />), it is closed and hence compact.
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A topological group that is the [[Projective limit|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.===
 
===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
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750508.png" /> be the set of integers larger than zero with the natural order relation, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p0750509.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505010.png" /> is the natural epimorphism, and put
+
$$
 
+
\tau _ {i}  ^ {j}  = \tau _ {i}  ^ {i+} 1 \tau _ {i+} 1  ^ {i+} 2 \dots \tau _ {j-} 1  ^ {j}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505011.png" /></td> </tr></table>
+
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505012.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505013.png" /> is the (additive) group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505015.png" />-adic integers.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505016.png" />-adic number field (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505017.png" />) is profinite as a topological group.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505018.png" /> be an abstract group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505019.png" /> be the family of its normal subgroups of finite index. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505020.png" /> one introduces the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505021.png" />, putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505022.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505023.png" />. This relation turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505024.png" /> into a pre-ordered directed set. Associate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505025.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505026.png" />, and to each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505028.png" />, the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505029.png" />. One obtains the profinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505030.png" />, called the profinite group completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505031.png" />. It is the separable completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505032.png" /> (cf. [[Separable completion of a ring|Separable completion of a ring]]) for the topology defined by the subgroups of finite index. The kernel of the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505033.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505034.png" />. The corresponding group is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505035.png" />, and is a [[Pro-p group|pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505036.png" />-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|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|pro- p $-
 +
group]].
  
4) Profinite groups naturally arise in Galois theory of (not necessarily finite) algebraic extensions of fields in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505037.png" /> be a [[Galois extension|Galois extension]] and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505038.png" /> is the family of all finite Galois extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505039.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505040.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505041.png" />, and one can introduce on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505042.png" /> the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505043.png" /> by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505044.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505045.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505046.png" /> then becomes pre-ordered. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505047.png" /> be the Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505048.png" />. To every pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505050.png" />, one naturally associates the homomorphism
+
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|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505051.png" /></td> </tr></table>
+
$$
 +
\tau _ {i}  ^ {j} :   \mathop{\rm Gal}  K _ {j} / k  \rightarrow  \mathop{\rm Gal}  K _ {i} / k .
 +
$$
  
The corresponding profinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505052.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505053.png" />, thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505054.png" /> can be considered as a profinite group. The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505055.png" /> forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075050/p07505056.png" /> a neighbourhood base of the identity (cf. [[Galois topological group|Galois topological group]]). This construction has a generalization in algebraic geometry in the definition of the fundamental group of a scheme.
+
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|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|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|Cohomology of groups]]; [[Galois cohomology|Galois cohomology]]) plays an important role in modern Galois theory.
 
A profinite group can be characterized as a compact totally-disconnected group (cf. [[Compact group|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|Cohomology of groups]]; [[Galois cohomology|Galois cohomology]]) plays an important role in modern Galois theory.

Revision as of 08:07, 6 June 2020


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 -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
How to Cite This Entry:
Profinite group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Profinite_group&oldid=48307
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article