Automorphism invariant Cartan subgroups and power maps of disconnected groups. (English) Zbl 1237.20040
The author extends a well-known result of R. Steinberg on the existence of an invariant maximal torus under a semisimple automorphism of an algebraic group over an algebraically closed field. It is shown that the same result holds when the underlying field is of characteristic zero, but not necessarily algebraically closed. This is applied to obtain the surjectivity of the power map \(g\mapsto g^n\) of disconnected algebraic groups of characteristic zero. A result of A. Borel on weak exponentiality in real Lie groups is also extended by relating it with the surjectivity of the square map.
Reviewer: L. N. Vaserstein (University Park)
MSC:
| 20G15 | Linear algebraic groups over arbitrary fields |
| 22E15 | General properties and structure of real Lie groups |
| 20G07 | Structure theory for linear algebraic groups |
Keywords:
Cartan subgroups of algebraic groups; power maps; surjectivity; weak exponentiality; real Lie groupsReferences:
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