On Cyclic Subgroups and the Conjugacy Problem

The conjugacy problem in three types of group constructions involving cyclic subgroups is discussed. First it is shown that if G has the solvable conjugacy problem and if $h \in G$ and $k \in G$ satisfy (a) h and k are not power conjugate to themselves or each other, (b) the power conjugacy problem in G with respect to h or k is solvable, and (c) the double coset solvability problem in G is solvable with respect to $\langle h\rangle$ and $\langle k\rangle$, then the HNN extension ${G^ \ast } = \langle G,t;{t^{ - 1}}ht = k\rangle$ has the solvable conjugacy problem. This result is used to deduce a similar theorem for free products with amalgamation, a fact first stated by Lipschutz. Then it is shown that if A and B are groups with the solvable conjugacy problem and $h \in A$ and $k \in B$ taken with themselves satisfy the conditions above in A and B, respectively, then $\langle A ^\ast B;[h,k] = 1\rangle$ has the solvable conjugacy problem.