5. Open Questions

1. Is any residually finite group with the maximum condition on subgroups a finite extension of a polycyclic group? (M. I. Kargapolov, [KN83, prob 1.31])
2. Is a finitely generated subgroup of a free soluble group finitely separable? (ie. are free soluble groups LERF? See Mal'cev's proof [Mal58] that soluble groups with the ascending chain condition on subgroups are ERF.) (M. I. Kargapolov, [KN83, prob 2.19])
3. Is there a finitely presented residually finite group with recursive, but not primitive recursive, solution to the word problem? (F. B. Cannonito, [KN83, prob 5.15])
4. Suppose G and H are finitely generated residually finite groups with the same set of finite homomorphic images. Are G and H isomorphic if one of them is free (or free soluble)? ( V. N. Remeslennikov, [KN83, prob 5.48])
5. Is every residually finite nonperiodic group satisfying the weak minimum condition for subgroups finite? (V. P. Shunkov, [KN83, prob 5.67])
6. Can a residually finite locally normal group be embedded in a Cartesian product of finite groups in such a way that each element of the group has at most a finite number of noncentral projections? (Yu. A. Gorchakov, [KN83, prob 6.8])
7. Suppose G is a residually finite Hopfian group and G^ is its profinite completion. Is G^ Hopfian (in the topological sense)? (O. V. Mel'nikov, [KN83, prob 6.30])
8. Which varieties B of groups have the property that G/B(G) is residually finite for every residually finite group G? (O. V. Mel'nikov, [KN83, prob 7.35])
9. Suppose that G is a residually finite group with one defining relation such that each subgroup of finite index also has one defining relation. Is G the fundamental group of a compact surface? (O. V. Mel'nikov, [KN83, prob 7.36])
10. Is every 1-relator group with torsion residually finite? (G. Baumslag, [GBaum73b, prob 13])
11. If G= and H= are residually finite, is I= also residually finite? (G. Baumslag, [GBaum73b, prob14])
12. If the derived group of a 1-relator (or, more generally, finitely generated) group G is locally free is G residually finite? (G. Baumslag, [GBaum73b, prob 16])
13. Let G={A*B|a=b} be the free product of two free groups A and B amalgamating a in A with b in B. If neither a nor b is a proper power, is G residually finite? (G. Baumslag, [GBaum73b, prob 18])
14. Is an extension of a finitely generated free group by the infinite cyclic group LERF? (G. P. Scott, [Gers87b, prob 5])
15. Given an equation W(x1,x2,...,a1,a2,...)=1 which has no solution in a free group F=<a1,a2,...>, is there a finite quotient of F in which w=1 has not solution? This property might be called "equational separability". For equations U(a1,a2,...)=1 it is residual finiteness, and for equations x-1U(a1,a2,...)x=V(a1,a2,...) it is conjugacy separability. (Leo Comerford, [Gers87b, prob 10])
16. Let C denote the class of all pc groups A such that for any pc group B, all free products with cyclic amalgamation A*ZB are pc. Is there a pc group not in class C? (Benny Evans, [Evan74])