Abstract. In this paper it is shown that if the Frattini subgroup of the fundamental group of a compact, orientable, irreducible, sufficiently large 3-manifold is nontrivial then the 3-manifold is a Seifert fibered space. We show further that the Frattini subgroup of the group of a Seifert fibered space is trivial or cyclic. As a corollary to our work we prove that every knot group has trivial Frattini subgroup.

Abstract.It is shown that residual finiteness is preserved by the generalized free product provided that the amalgamated subgroups are retracts of their respective factors. This result is applied to knot groups. The outcome is that the question of residual finiteness for knot groups need only be answered for prime knots.

It is shown that residual finiteness is preserved by the generalized free product provided that the amalgamated subgroups are retracts of their respective factors. This result is applied to knot groups. The outcome is that the question of residual finiteness for knot groups need only be answered for prime knots.

Let G be a finitely generated metabelian group whose derived group G′ has finite rank. It is shown that G can be embedded in a finitely presented metabelian group H with H′ of finite rank.

Abstract.It is shown that each finitely generated torsion-free abelian-bycyclic group has solvable conjugacy problem. This is done by showing that solving the conjugacy problem for these groups is equivalent to a certain decision problem for modules over the complex group algebra of an infinite cyclic group.