Residual property (mathematics): Difference between revisions

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In the [[mathematics|mathematical]] field of [[group theory]], a group is '''residually ''X''''' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''".
In the [[mathematics|mathematical]] field of [[group theory]], a group is '''residually ''X''''' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''".


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==References==
==References==
* {{cite book | title=The theory of groups | author=Marshall Hall jr | authorlink=Marshall Hall (mathematician) | location=New York | publisher=Macmillan | year=1959 | page=16 }}
* {{cite book | title=The theory of groups | author=Marshall Hall jr | authorlink=Marshall Hall (mathematician) | location=New York | publisher=Macmillan | year=1959 | page=16 }}
[[Category:Infinite group theory]]
[[Category:Properties of groups]]
{{algebra-stub}}

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In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".

Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that .

More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of where H is a group with property X.

Examples

Important examples include:

References