# 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 $\operatorname{RF}\rho$; 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In outdated terminology a residually-finite group is called a finitely-approximated group, which is also the word-for-word translation of the Russian for this notion.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) |

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Residually-finite group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Residually-finite_group&oldid=31730