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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On cyclic subgroups and the conjugacy problem


Author: R. Daniel Hurwitz
Journal: Proc. Amer. Math. Soc. 79 (1980), 1-8
MSC: Primary 20F10; Secondary 03D40
MathSciNet review: 560573
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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 \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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0560573-2
PII: S 0002-9939(1980)0560573-2
Article copyright: © Copyright 1980 American Mathematical Society