On the Root-Class Residuallity of Generalized Free Products

Let K be a root class of groups. It is proved that a free product of any family of residually K groups with one amalgamated subgroup, which is a retract in all free factors, is residually K. The sufficient condition for a generalized free product of two groups to be residually K is also obtained, provided that the amalgamated subgroup is normal in one of the free factors and is a retract in another.

On the residual solvability of generalized free products of solvable groups

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience In this paper, we study the residual solvability of the generalized free product of solvable groups.

Near Frattini subgroups of residually finite generalized free products of groups

LetG=A★HBbe the generalized free product of the groupsAandBwith the amalgamated subgroupH. Also, letλ(G)andψ(G)represent the lower near Frattini subgroup and the near Frattini subgroup ofG, respectively. IfGis finitely generated and residually finite, then we show thatψ(G)≤H, providedHsatisfies a nontrivial identical relation. Also, we prove that ifGis residually finite, thenλ(G)≤H, provided: (i)Hsatisfies a nontrivial identical relation andA,Bpossess proper subgroupsA1,B1of finite index containingH; (ii) neitherAnorBlies in the variety generated byH; (iii)H<A1≤AandH<B1≤B, whereA1andB1each satisfies a nontrivial identical relation; (iv)His nilpotent.

Cyclic Subgroup Separability of Generalized Free Products

We derive a criterion for a generalized free product of groups to be cyclic subgroup separable. We see that most of the known results for cyclic subgroup separability are covered by this criterion, and we apply the criterion to polygonal products of groups. We show that a polygonal product of finitely generated abelian groups, amalgamating cyclic subgroups, is cyclic subgroup separable.

The solutions to two problems on permutational products

The two problem, both raised in the literature, are: (I) Is there, amongst all the permutational products (p.p.s.) on the amalgam = (A, B; H) at least one which is a minimal generalized regular product? (II) If one of the p.p.s. on is isomorphic to the generalized free product (g.f.p.) F on U are they all? We answer both of them negatively.

Notes on Frattini Subgroups of Generalized Free Products with Cyclic Amalgamation

The problem of the exact location of the Frattini subgroup 4>(G) of a generalized free product G = (A*B)H was first raised by Higman and Neumann [5]. Solutions to special cases of the problem can be found in [1], [2], [8], [9] and [10]. The purpose of this note is to extend the results of [2], [8], and to simplify the proof of Whittemore's theorem [10]. We also apply our result to give simple proofs of certain classes of knot groups that have trivial Frattini subgroups. The proof that every knot group has trivial Frattini subgroup hard and long (footnote 2, p. 56).

Fuchsian Subgroups of the Picard Group

The Picard group Γ = PSL2 (Z(i)) is the group of linear transformationswith a, b, c, d Gaussian integers.Γ is of interest both as an abstract group and in automorphic function theory [10]. In [10] Waldinger constructed a subgroup H of finite index which is a generalized free product, while in [1] Fine showed that T is a semidirect product with the subgroup H, contained as a subgroup of finite index in the normal factor.