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Information AboutProfinite Group |
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DEFINITION Formally, a pro-finite group is a Hausdorff Compact and Totally Disconnected Topological Group . Equivalently, one can define a pro-finite groups as topological groups Isomorphic to Inverse Limits (in the Category of topological groups) of an inverse system of Finite Group s, regarded as Discrete topological groups. EXAMPLES
PROPERTIES AND FACTS Every Product of (arbitrarily many) pro-finite groups is pro-finite; the topology arising from the pro-finiteness agrees with the Product Topology . Every Closed subgroup of a pro-finite group is itself pro-finite; the topology arising from the pro-finiteness agrees with the Subspace Topology . If ''N'' is a closed normal subgroup of a pro-finite group ''G'', then the Factor Group ''G''/''N'' is pro-finite; the topology arising from the pro-finiteness agrees with the Quotient Topology . Since every pro-finite group ''G'' is compact Hausdorff, we have a Haar Measure on ''G'', which allows us to measure the "size" of subsets of ''G'', compute certain probabilities, and integrate functions on ''G''. PRO-FINITE COMPLETION Given an arbitrary group ''G'', there is a related pro-finite group ''G''^, the pro-finite completion of ''G''. It is defined as the inverse limit of the groups ''G''/''N'', where ''N'' runs through the group homomorphism ''g'' : ''G''^ → ''H'' with ''f'' = ''g''η. IND-FINITE GROUPS There is a notion of ind-finite group, which is the concept Subgroup is finite. This is equivalent, in fact, to being 'ind-finite'. By applying Pontryagin Duality , one can see that Abelian pro-finite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian Torsion Group s. See also: Locally Cyclic Group . FURTHER READING
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