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Transactions of the American Mathematical Society

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

 

 

Non-Hopfian groups with fully invariant kernels. I


Author: Michael Anshel
Journal: Trans. Amer. Math. Soc. 170 (1972), 231-237
MSC: Primary 20F05
DOI: https://doi.org/10.1090/S0002-9947-1972-0304491-3
MathSciNet review: 0304491
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Abstract: Let $ \mathcal{L}$ consist of the groups $ G(l,m) = (a,b;{a^{ - 1}}{b^l}a = {b^m})$ where $ \vert l\vert \ne 1 \ne \vert m\vert,lm \ne 0$ and $ l,m$ are coprime. We characterize the endomorphisms of these groups, compute the centralizers of special elements and show that the endomorphism $ a \to a,b \to {b^l}$ is onto with a nontrivial fully invariant kernel. Hence $ G(l,m)$ is non-Hopfian in the'fully invariant sense.'


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DOI: https://doi.org/10.1090/S0002-9947-1972-0304491-3
Keywords: Groups with one defining relation, endomorphisms of groups, fully invariant subgroups, characteristic subgroups, reduced free groups, relatively free groups
Article copyright: © Copyright 1972 American Mathematical Society