TWO PROBLEMS FOR SOLVABLE AND NILPOTENT GROUPS

Section 1 gives a brief review of known results on embeddings of solvable, nilpotent, and polycyclic groups in 2-generated groups from these classes, including the description of the author's recently obtained solution to the Mikaelian-Ol'schanskii problem on embeddings of finitely generated solvable groups of derived length l in solvable groups of derived length l + 1 with a fixed small number of generators. Section 2 contains a somewhat more extensive review of known results on the rational subset membership problem for groups, including the presentation of the author's recently obtained solution to the Laurie-Steinberg-Kambites-Silva-Zetsche problem of whether the membership problem is decidable for finitely generated submonoids of free nilpotent groups.