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In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every nontrivial element ''g'' in ''G'' there is a homomorphism ''h'' from ''G'' to a finite group, such that : There are a number of equivalent definitions: *A group is residually finite if for each non-identity element in the group, there is a normal subgroup of finite index not containing that element. *A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial. *A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial. *A group is residually finite if and only if it can be embedded inside the direct product of a family of finite groups. == Examples == Examples of groups that are residually finite are finite groups, free groups, finitely generated nilpotent groups, polycyclic-by-finite groups, finitely generated linear groups, and fundamental groups of 3-manifolds. Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any inverse limit of residually finite groups is residually finite. In particular, all profinite groups are residually finite. Counterexamples can be given using the fact that all finitely generated resiually finite groups are Hopfian groups, so for example the Baumslag–Solitar group ''B''(2,3) is not Hopfian. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Residually finite group」の詳細全文を読む スポンサード リンク
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