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

DOI:
https://doi.org/10.1090/S0002-9939-1980-0560573-2

MathSciNet review:
560573

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 and 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 and , then the HNN extension 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 and taken with themselves satisfy the conditions above in *A* and *B*, respectively, then has the solvable conjugacy problem.

**[1]**M. Anshel,*Conjugate powers in HNN groups*, Proc. Amer. Math. Soc.**54**(1976), 19-23. MR**52**#14059. MR**0393249 (52:14059)****[2]**M. Anshel and P. Stebe,*The solvability of the conjugacy problem for certain HNN groups*, Bull. Amer. Math. Soc.**80**(1974), 266-270. MR**0419615 (54:7633)****[3]**D. J. Collins,*On recursively enumerable degrees and the conjugacy problem*, Acta Math.**122**(1969), 115-160. MR**39**#4001. MR**0242671 (39:4001)****[4]**L. P. Comerford and B. Truffault,*The conjugacy problem for free products of sixth-groups with cyclic amalgamation*, Math. Z.**149**(1976), 169-181. MR**53**#13418. MR**0409666 (53:13418)****[5]**R. D. Hurwitz,*On the conjugacy problem in certain product groups*, Ph. D. Thesis, University of Illinois, Urbana, 1974.**[6]**-,*On the conjugacy problem in a free product with commuting subgroups*, Math. Ann.**221**(1976), 1-8. MR**54**#414. MR**0412288 (54:414)****[7]**S. Lipschutz,*Generalization of Dehn's result on the conjugacy problem*, Proc. Amer. Math. Soc.**17**(1966), 759-762. MR**33**#5706. MR**0197541 (33:5706)****[8]**-,*The conjugacy problem and cyclic amalgamations*, Bull. Amer. Math. Soc.**81**(1975), 114-116. MR**52**#580. MR**0379675 (52:580)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
20F10,
03D40

Retrieve articles in all journals with MSC: 20F10, 03D40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0560573-2

Article copyright:
© Copyright 1980
American Mathematical Society