Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Conjugacy separability of groups of integer matrices

Author: Peter F. Stebe
Journal: Proc. Amer. Math. Soc. 32 (1972), 1-7
MSC: Primary 20.75
MathSciNet review: 0289666
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] H. Bass, M. Lazard and J.-P. Serre, Sous-groupes d'indice fini dans $ {\text{SL}}(n,Z)$, Bull. Amer. Math. Soc. 70 (1964), 385-392. MR 28 #5117. MR 0161913 (28:5117)
  • [2] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 2nd. ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 14, Springer-Verlag, Berlin, 1965. MR 30 #4818. MR 0174618 (30:4818)
  • [3] W. Magnus, A. Karass and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Pure and Appl. Math., vol. 13, Interscience, New York, 1966. MR 34 #7617.
  • [4] A. W. Mostowski, On the decidability of some problems in special classes of groups, Fund. Math. 59 (1966), 123-135. MR 37 #292. MR 0224693 (37:292)
  • [5] B. H. Neumann, An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London, Ser. A. 246 (1954), 503-554. MR 16, 10. MR 0062741 (16:10d)
  • [6] P. F. Stebe, A residual property of certain groups, Proc. Amer. Math. Soc. 26 (1970), 37-42. MR 41 #5494. MR 0260874 (41:5494)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20.75

Retrieve articles in all journals with MSC: 20.75

Additional Information

Keywords: Group, unimodular group, conjugacy problem, conjugacy separable
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society