On HNN-extensions in the class of groups of large odd exponent

Abstract

A sufficient condition for the existence of HNN-extensions in the class of groups of odd exponent $n\gg 1$ is given in the following form. Let $Q$ be a group of odd exponent $n>2^{48}$ and $\mathcal{G}$ be an HNN-extension of $Q$. If $A\in \mathcal{G}$ then let $\mathcal{F}(A)$ denote the maximal subgroup of $Q$ which is normalized by $A$. By ${\tau }_A$ denote the automorphism of $\mathcal{F}(A)$ which is induced by conjugation by $A$. Suppose that for every $A\in \mathcal{G}$, which is not conjugate to an element of $Q$, the group $<{\tau }_A,\mathcal{F}(A)>$ has exponent $n$ and, in addition, equalities $A^{-k}{q}_0{A}^k=q_k$, where $q_k\in Q$ and $k=0,1,...,[2^{-16}n]$ ($[2^{-16}n]$ is the integer part of $2^{-16}n$), imply that $q_0\in \mathcal{F}(A)$. Then the group $Q$ naturally embeds in the quotient $\mathcal{G}/{\mathcal{G}}^n$, that is, there exists an analog of the HNN-extension $\mathcal{G}$ of $Q$ in the class of groups of exponent $n$.

11 pages