On Hnn-extensions in the Class of Groups of Large Odd Exponent

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, which is not conjugate to an element of Q, the group 〈τA,F(A)〉 has exponent n and, in addition, equalities A q0A = qk, where qk ∈ Q and k = 0, 1, . . . , [2 n] ([2n] is the integer part of 2n), imply that q0 ∈ F(A). Then the group Q naturally embeds in the quotient G/G, that is, there exists an analog of the HNN-extension G of Q in the class of groups of exponent n.