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A commutativity criterion for closed subgroups of compact Lie groups.

Author: Joseph A. Wolf
Journal: Proc. Amer. Math. Soc. 27 (1971), 619-622
MSC: Primary 22.50; Secondary 17.00
MathSciNet review: 0277663
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Abstract: Let $ \Gamma $ be a closed subgroup of a compact Lie group $ G$. If the identity component $ {\Gamma _0}$ is commutative, and if the order of $ \Gamma /{\Gamma _0}$ is prime to the order of the Weyl group of $ G$, then it is shown that $ \Gamma $ is commutative. If $ G$ is a classical group this extends a theorem of Burnside on finite linear groups. If $ G$ is exceptional this gives some information on Cayley-Dickson algebras, Jordan algebras and the Cayley protective plane.

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

  • [1] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029. MR 29 #3538. MR 0166261 (29:3538)
  • [2] J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967. MR 36 #829. MR 0217740 (36:829)

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Keywords: Lie group, closed subgroup, commutative subgroup, Weyl group, linear group, Jordan algebra
Article copyright: © Copyright 1971 American Mathematical Society

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