Residually-finite group
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In outdated terminology a residually-finite group is called a finitely-approximated group, which is also the word-for-word translation of the Russian for this notion.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) |
Residually-finite group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residually-finite_group&oldid=31730