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Conjugacy separability of groups of integer matrices


Author: Peter F. Stebe
Journal: Proc. Amer. Math. Soc. 32 (1972), 1-7
MSC: Primary 20.75
DOI: https://doi.org/10.1090/S0002-9939-1972-0289666-X
MathSciNet review: 0289666
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Abstract: An element g of a group G is conjugacy distinguished if and only if given any element h of G either g is conjugate to h or there is a homomorphism $ \xi $ of G onto a finite group such that $ \xi (g)$ is not conjugate to $ \xi (h)$. Following A. W. Mostowski, a group is conjugacy separable if every one of its elements is conjugacy distinguished. Let $ {\text{GL}}(n,Z)$ be the group of $ n \times n$ integer matrices with determinant $ \pm 1$. Let $ {\text{SL}}(n,Z)$ be the subgroup of $ {\text{GL}}(n,Z)$ consisting of matrices with determinant $ + 1$. It is shown that $ {\text{GL}}(n,Z)$ and $ {\text{SL}}(n,Z)$ are conjugacy separable if and only if $ n = 1$ or 2. The groups $ {\text{SL}}(n,Z)$ are also called unimodular groups. Let $ {\text{GL}}(n,{Z_p})$ be the group of invertible p-adic integer matrices and $ {\text{SL}}(n,{Z_p})$ be the group of p-adic integer matrices with determinant 1. It is shown that $ {\text{GL}}(n,{Z_p})$ and $ {\text{SL}}(n,{Z_p})$ are conjugacy separable for all n and all p.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0289666-X
Keywords: Group, unimodular group, conjugacy problem, conjugacy separable
Article copyright: © Copyright 1972 American Mathematical Society

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