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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A residual property of certain groups


Author: P. F. Stebe
Journal: Proc. Amer. Math. Soc. 26 (1970), 37-42
MSC: Primary 20.48
MathSciNet review: 0260874
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Abstract: An element $ a$ of a group $ G$ is called conjugacy distinguished or c.d. in $ G$ if and only if given any element $ b$ of $ G$ either $ a$ is conjugate to $ b$ or there is a homomorphism $ \xi $ from $ G$ onto a finite group $ H$ such that $ \xi (a)$ and $ \xi (b)$ are not conjugate in $ H$. Following A. Mostowski, a group $ G$ is called conjugacy separable or c.s. if every element of $ G$ is c.d. A. Mostowski remarks that a direct product of c.s. groups is c.s. and proves that the conjugacy problem can be solved for finitely presented c.s. groups. N. Blackburn proves that finitely generated nilpotent groups are c.s. In this paper it is proven that a free product of c.s. groups is c.s., a free group is c.s. and that every element of infinite order in a finite extension of a free group is c.d.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0260874-5
Keywords: Group, c. s. group, residual property, conjugacy problem
Article copyright: © Copyright 1970 American Mathematical Society