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Residually-finite group

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A group that can be approximated by finite groups. Let be a group and \rho a relation (in other words, a predicate) between elements and sets of elements, defined on G and all homomorphic images of it (for example, the binary relation of equality of elements, the binary relation "the element x belongs to the subgroup y" , the binary relation of conjugacy of elements, etc.). Let K be a class of groups. One says that G can be approximated by groups in K relative to \rho (or: G is residual in K relative to \rho) if for any elements and sets of elements of G that are not in relation \rho there is a homomorphism of G onto a group in K under which the images of these elements and sets are also not in relation \rho. Approximability relative to the relation of equality of elements is simply called approximability. A group can be approximated by groups in a class K if and only if it is contained in a Cartesian product of groups in K. Residual finiteness relative to \rho is denoted by \operatorname{RF}\rho; in particular, if \rho runs through the predicates of equality, conjugacy, belonging to a subgroup, belonging to a finitely-generated subgroup, etc., then one obtains the properties (and classes) \operatorname{RF}E, \operatorname{RF}C, \operatorname{RF}B, \operatorname{RF}B_\omega, etc. The presence of these properties in a group implies the solvability of the corresponding algorithmic problem.

References

[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)


Comments

In outdated terminology a residually-finite group is called a finitely-approximated group, which is also the word-for-word translation of the Russian for this notion.

For a fuller account on residually-finite groups see [a1].

References

[a1] D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
How to Cite This Entry:
Residually-finite group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residually-finite_group&oldid=31730
This article was adapted from an original article by Yu.I. Merzlyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article