Profinite completion of the integers: Difference between revisions

Revision as of 00:26, 15 January 2012

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The profinite completion of the integers is defined in the following equivalent ways. Note that these definitions can be interpreted both as defining the group abstractly and as defining the group as a topological group:

The group obtained by taking the profinite completion of the group of integers (viewed as a discrete group).
The external direct product, over all prime numbers $p$ , of the additive group of p-adic integers for the prime $p$ . Note that we take the product topology from the topologies on these groups.

Arithmetic functions

Function	Value	Explanation
order of a profinite group	$p^{\infty}$ for that prime $p$ .	This is a supernatural number as is the convention for orders of profinite groups.

Group properties

Abstract group properties

Property	Satisfied?	Explanation
abelian group
aperiodic group

Topological group properties

Here, the topology is from the profinite group structure.

Property	Satisfied?	Explanation
profinite group	Yes	By definition
compact group	Yes	profinite groups are compact
T0 topological group	Yes	profinite groups are Hausdorff, hence $T_{0}$
totally disconnected group	Yes	profinite groups are totally disconnected
topologically cyclic group	Yes	$Z$ , the subgroup generated by the element that projects to 1 mod $p^{n}$ for all $n$ , is a dense subgroup. In fact, this group can be thought of as a compactification of $Z$ .
topologically finitely generated group	Yes	Follows from being topologically cyclic.
connected topological group	No	It is nontrivial and totally disconnected.

@@ Line 7: / Line 7: @@
 # The group obtained by taking the [[defining ingredient::profinite completion]] of the [[group of integers]] (viewed as a discrete group).
 # The [[external direct product]], over all [[prime number]]s <math>p</math>, of the additive [[group of p-adic integers]] for the prime <math>p</math>. Note that we take the product topology from the topologies on these groups.
+==Arithmetic functions==
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| [[order of a profinite group]] || <math>p^\infty</math> for that prime <math>p</math>. || This is a [[supernatural number]] as is the convention for orders of profinite groups.
+|}
+==Group properties==
+===Abstract group properties===
+{| class="sortable" border="1"
+! Property !! Satisfied? !! Explanation
+|-
+| [[satisfies property::abelian group]] || ||
+|-
+| [[satisfies property::aperiodic group]] || ||
+|}
+===Topological group properties===
+Here, the topology is from the profinite group structure.
+{| class="sortable" border="1"
+! Property !! Satisfied? !! Explanation
+|-
+| [[satisfies property::profinite group]] ||Yes || By definition
+|-
+| [[satisfies property::compact group]] || Yes || profinite groups are compact
+|-
+| [[satisfies property::T0 topological group]] || Yes || profinite groups are Hausdorff, hence <math>T_0</math>
+|-
+| [[satisfies property::totally disconnected group]] || Yes ||profinite groups are totally disconnected
+|-
+| [[satisfies property::topologically cyclic group]] || Yes || <math>\mathbb{Z}</math>, the subgroup generated by the element that projects to 1 mod <math>p^n</math> for all <math>n</math>, is a dense subgroup. In fact, this group can be thought of as a compactification of <math>\mathbb{Z}</math>.
+|-
+| [[satisfies property::topologically finitely generated group]] || Yes || Follows from being topologically cyclic.
+|-
+| [[dissatisfies property::connected topological group]] || No || It is nontrivial and totally disconnected.
+|}