The Free Product of Two Groups with a Malnormal Amalgamated Subgroup

Abstract

In [ 1 ], B. Baumslag defined a subgroup U of a group G to be malnormal in G if gug –1 ∈ U, 1 ≠ u ∈ U , implies that g ∈ U. Baumslag considered the class of amalgamated products ( A * B; U ) in which U is malnormal in both A and B. These amalgamated products play an important role in the investigations of B. B. Newman [ 13 ] of groups with one defining relation having torsion. In this paper, we shall be concerned primarily with a generalization of this class. Let U be a subgroup of a group G and let u ∈ U. Then the extended normalizer E G (u, U) of u relative to U in G is defined by if u ≠ 1, and by E G (u, U) = U if u = 1.