Residually-finite group

A group that can be approximated by finite groups. Let $G$ 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 $\mathrm{RF}\rho$; 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) $\mathrm{RF}E$, $\mathrm{RF}C$, $\mathrm{RF}B$, $\mathrm{RF}{B}_{\omega }$, etc. The presence of these properties in a group implies the solvability of the corresponding algorithmic problem.

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In outdated terminology a residually-finite group is called a finitely-approximated group, which is also the word-for-word translation of the Russian for this notion.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

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