On cyclic subgroups and the conjugacy problem

Abstract

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 ∈ G h \in G and k ∈ G 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 ⟨ h ⟩ \langle h\rangle and ⟨ k ⟩ \langle k\rangle , then the HNN extension G ∗ = ⟨ G , t ; t − 1 h t = k ⟩ {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 ∈ A h \in A and k ∈ B k \in B taken with themselves satisfy the conditions above in A and B, respectively, then ⟨ A ∗ B ; [ h , k ] = 1 ⟩ \langle A ^\ast B;[h,k] = 1\rangle has the solvable conjugacy problem.