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Residual property (mathematics)
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Residual property (mathematics) : ウィキペディア英語版
Residual property (mathematics)
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 h(g)\neq e.
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 \phi\colon G \to H where ''H'' is a group with property ''X''.
==Examples==
Important examples include:
* Residually finite
* Residually nilpotent
* Residually solvable
* Residually free

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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