On the Root-Class Residuality of Certain Free Products of Groups with Normal Amalgamated Subgroups

Let Kbe a root class of groups closed under taking quotient groups, G be a free product of groups A and B with amalgamated subgroups H and K. Let also H be normal in A, K be normal in B, and Aut(G)(H) denote the set of automorphisms of H induced by all inner automorphisms of G. We prove a criterion for G to be residually a K-group provided Aut(G)(H) is an abelian group or it satisfies some other conditions. We apply this result in the cases when A and B are bounded nilpotent groups or A/H, B/K is an element of K.