[2] All three problems, formulated as ranging over all finitely presented groups, are undecidable.
In fact the problem of deciding whether a finitely presented group is trivial is undecidable,[3] a consequence of the Adian–Rabin theorem due to Sergei Adian and Michael O. Rabin.
However, there are some classes of finitely presented groups for which the restriction of the isomorphism problem is known to be decidable.
[8] Both have quasi-polynomial-time algorithms, the former since 1978 attributed to Robert Tarjan[9] and the latter since 2015 by László Babai.
[10] A small but important improvement for the case p-groups of class 2 was obtained in 2023 by Xiaorui Sun.