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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Some one-relator Hopfian groups


Author: Donald J. Collins
Journal: Trans. Amer. Math. Soc. 235 (1978), 363-374
MSC: Primary 20E30
DOI: https://doi.org/10.1090/S0002-9947-1978-0460468-4
MathSciNet review: 0460468
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Abstract: The group presented by

$\displaystyle (a,t;{t^{ - 1}}{a^l}t = {a^m})$

is non-Hopfian if $ l,m \ne \pm 1$ and $ \pi (l) \ne \pi (m)$, where $ \pi (l)$ and $ \pi (m)$ denote the sets of prime divisors of l and m. By contrast, we prove that if w is a word of the free group $ F({a_1},{a_2})$ which is not primitive and not a proper power, then the group $ ({a_1},{a_2},t;{t^{ - 1}}{w^l}t = {w^m})$ is Hopfian.

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DOI: https://doi.org/10.1090/S0002-9947-1978-0460468-4
Article copyright: © Copyright 1978 American Mathematical Society