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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Products of similar matrices


Author: Dave Witte
Journal: Proc. Amer. Math. Soc. 126 (1998), 1005-1015
MSC (1991): Primary 06F15, 20G15, 20G25; Secondary 20F99, 20H05
MathSciNet review: 1451837
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Abstract: Let $A$ and $B$ be $n \times n$ matrices of determinant $1$ over a field $K$, with $n >2$ or $|K|>3$. We show that if $A$ is not a scalar matrix, then $B$ is a product of matrices similar to $A$. Analogously, we conjecture that if $a$ and $b$ are elements of a semisimple algebraic group $G$ over a field of characteristic zero, and if there is no normal subgroup of $G$ containing $a$ but not $b$, then $b$ is a product of conjugates of $a$. The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. Thus, in a connected, semisimple Lie group with finite center, the only conjugation-invariant subsemigroups are the normal subgroups.


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Additional Information

Dave Witte
Email: dwitte@math.okstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04368-8
PII: S 0002-9939(98)04368-8
Received by editor(s): August 2, 1996
Received by editor(s) in revised form: October 1, 1996
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1998 American Mathematical Society