Let $G$ be a finitely presented simple group. By Kuznetsov (1958), $G$ has decidable word problem. However, by Scott [1], $G$ may have undecidable conjugacy problem. Is anything known about other decision problems for $G$, in generality? I am particularly interested in the question of the subgroup membership problem, i.e. the problem of, given a set of words $w\_1, \dots, w\_k$ and a word $w$, deciding whether or not $w$ belongs to the subgroup $\langle w\_1, \dots, w\_k \rangle$ of $G$. This is undecidable already for the direct product $H \times H$ of any group $H$ admitting a finitely presented quotient with undecidable word problem (in particular, we can take $H$ to be a free group of rank $2$), but such direct products are of course very far from being simple.


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[1] *Scott, Elizabeth A.*, [**A finitely presented simple group with unsolvable conjugacy problem**](https://doi.org/10.1016/0021-8693(84)90174-1), J. Algebra 90, 333-353 (1984). [ZBL0544.20029](https://zbmath.org/?q=an:0544.20029).



Subgroup membership problem in simple groups
As another example, the problem of computing the order of an element of the finitely presented simple Brin–Thompson group $2V$ is undecidable by


*Belk, James; Bleak, Collin*, [**Some undecidability results for asynchronous transducers and the Brin-Thompson group (2V)**](https://doi.org/10.1090/tran/6963), Trans. Am. Math. Soc. 369, No. 5, 3157-3172 (2017). [ZBL1364.20015](https://zbmath.org/?q=an:1364.20015).


