Suppose that G is a group, H and K are proper isomorphic central subgroups of G, an HNN-extension with the associated K. We prove necessary sufficient conditions for to be residually C-group, where C class groups closed under taking subgroups, extensions, homomorphic images, Cartesian products form ∏y∈YXy, X,Y∈C Xy copy X each y∈Y.

A sufficient condition for the existence of HNN-extensions in the class of groups of odd exponent n ≫ 1 is given in the following form. Let Q be a group of odd exponent n > 2 and G be an HNN-extension of Q. If A ∈ G then let F(A) denote the maximal subgroup of Q which is normalized by A. By τA denote the automorphism of F(A) which is induced by conjugation by A. Suppose that for every A ∈ G, wh...