See also METABELIAN GROUPS and NILPOTENT GROUPS.
 

(S1) (A.I.Mal'cev)  Describe the automorphism group of a free solvable group of finite rank.  In particular, is this group finitely generated? Background

(S2) (M.I.Kargapolov) The word problem for groups admitting a single defining relation in the variety of all solvable groups of a given derived length.   Background

(S3) (M.I.Kargapolov) Is it true that every group of rank > 2 admitting a single defining relation in the variety of all solvable groups of a given derived length, has trivial centre?  Background

(S4) (M.I.Kargapolov) Is there a number N = N(k,d) so that every element of the commutator subgroup of a free solvable group of rank k and derived length d, is a product of N commutators?  Background

(S5) (P.M.Neumann)  Is it true that if A, B are finitely generated solvable Hopfian groups, then  AxB  is Hopfian?

(S6) (V.N.Remeslennikov) The conjugacy problem for finitely generated abelian-by-polycyclic groups.

(S7) (D.Robinson) Is there a finitely presented solvable group satisfying the maximum condition on normal subgroups, with unsolvable word problem ?

(S8) (G.Baumslag, V.Remeslennikov) Is a finitely generated free solvable group of derived length 3 embeddable in a finitely presented solvable group?

*(S9) (B.Fine, V.Shpilrain) Let  u  be an element of a group  G.  We call  u  a test element if, whenever  f(u)=u  for some endomorphism  f  of the group G,  this   f  is actually an automorphism of G.  Does the free  solvable group of rank 2 and derived length d>2  have any test elements?   Background