Is It possible to determine whether the given finitely presented group is residually finite with MAGMA or GAP? I am working on finitely presented groups with more than 5 generators and relators and I'm so curious: is it possible to determine residually finitness of finitely presented groups with MAGMA or GAP?
 A: To precise Giles Gardam's answer, let me add the following.
The Adian-Rabin theorem shows that residual finiteness is undecidable, by showing that no algorithm can stop exactly on non-residually finite groups.
In other words, the set of non-residually finite groups is not recursively enumerable -or not semi-decidable.
However, the Adian-Rabin theorem fails to prove that there cannot exist an algorithm that stops exactly on finite presentations of residually finite groups, i.e. an algorithm that will testify that a given group is residually finite and not stop otherwise.
Yet this is also the case, it follows easily from the article Algorithmically complex residually finite groups of Kharlampovich, Myasnikov and Sapir, see Corollary 22 of my article for a written out proof.
This result contrasts for instance with the situation for hyperbolic groups, a property that the Adian-Rabin theorem also proves undecidable, but which was shown to be partially decidable by Papasoglu.
A: It is a consequence of the Adian-Rabin theorem that there is no algorithm that decides, given a finite presentation of a group, whether the group is residually finite.
In a similar spirit, it is undecidable whether a finitely presented group has any non-trivial finite quotients, by
Bridson, Martin R.; Wilton, Henry, The triviality problem for profinite completions, Invent. Math. 202, No. 2, 839-874 (2015). ZBL1360.20020.
