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Matrix Groups pp 106-121 | Cite as

Covering by Maximal Tori

  • Morton L. Curtis
Part of the Universitext book series (UTX)

Abstract

In Exercise 4 of Chapter VII one showed that if T is a maximal torus in a matrix group G , then for any x ∈ G , xTx -1 is also a maximal torus. What we prove In this chapter is that if T is our standard maximal torus in one of our connected matrix groups G , then (†)
$$G = \mathop \cup \limits_{X \in G} {\text{ }}xT{x^{ - 1}}$$
showing that every element of G lies in at least one maximal torus. To say that \(G = \mathop \cup \limits_{X \in G} {\text{ }}xT{x^{ - 1}}\) is to say that given y ∈ G there exists x∈G such that
$$y \in xT{x^{ - 1}}$$
.

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Copyright information

© Springer-Verlag New York Inc. 1984

Authors and Affiliations

  • Morton L. Curtis
    • 1
  1. 1.Department of MathematicsRice University, Weiss School of Natural SciencesHoustonUSA

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