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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''". |
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| | Formally, a group ''G'' is residually ''X'' if for every non-trivial element ''g'' there is a [[Group homomorphism|homomorphism]] ''h'' from ''G'' to a group with property ''X'' such that <math>h(g)\neq e</math>. |
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| | 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 <math>\phi\colon G \to H</math> where ''H'' is a group with property ''X''. |
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| | ==Examples== |
| | Important examples include: |
| | * [[Residually finite group|Residually finite]] |
| | * Residually [[nilpotent group|nilpotent]] |
| | * Residually [[solvable group|solvable]] |
| | * Residually [[free group|free]] |
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| | ==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 }} |
| | |
| | [[Category:Infinite group theory]] |
| | [[Category:Properties of groups]] |
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| | {{Abstract-algebra-stub}} |
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
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