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 }} | ||
Revision as of 14:07, 28 October 2008
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:
- Residually finite
- Residually nilpotent
References
- Marshall Hall jr (1959). The theory of groups. New York: Macmillan.