5-reflectionality of anisotropic orthogonal groups over valuation rings. (English) Zbl 0747.20026
Let \(R\) be a valuation ring with maximal ideal \(J\), and let \(V\) be a free \(R\)-module of finite rank \(n\) equipped with a symmetric bilinear form whose induced form on \(V/JV\) is anisotropic. The author shows that if \(\pi\) is an element of the orthogonal group of \(V\) such that the image of \(\pi^ 2-1\) is a direct summand of \(V\), then \(\pi\) is a product of two orthogonal involutions. Further, if \(n\) is odd, every orthogonal mapping of \(V\) is a product of five involutions; four suffice when \(n\) is even. A new proof of D. Ljubić’s theorem [Can. Math. Bull. 32, 54-63 (1989; Zbl 0635.51010)] on isometry group of absolute geometries is also presented.
Reviewer: W.H.Gustafson (Lubbock)
MSC:
20H25 | Other matrix groups over rings |
20F05 | Generators, relations, and presentations of groups |
15A23 | Factorization of matrices |
11E57 | Classical groups |
Keywords:
anisotropic form; valuation ring; symmetric bilinear form; orthogonal group; product of two orthogonal involutions; orthogonal mapping; product of five involutions; isometry groupReferences:
[1] | F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff, 2^nd edition, Springer 1973. · Zbl 0254.50001 |
[2] | Dokovic, D. Z., The Product of Two Involutions in the Unitary Group of a Hermitian Form, Indiana Univ. Math. J., 21, 449-456 (1971) · Zbl 0226.20036 · doi:10.1512/iumj.1971.21.21035 |
[3] | Ellers, E. W.; Frank, R., Products of Quasireflections and Transvections over Local Rings, J. Geom., 31, 69-78 (1988) · Zbl 0639.20031 · doi:10.1007/BF01222386 |
[4] | H. Kinder, Begründung dern-dimensionalen absoluten, Geometrie aus dem Spiegelungsprinzip, Dissertation, Kiel 1965. |
[5] | Knüppel, F.; Nielsen, K., Products of Involutions in O^+(V), Lin. Alg. and Appl., 94, 217-222 (1987) · Zbl 0623.20036 · doi:10.1016/0024-3795(87)90092-9 |
[6] | Lubić, D., Bireflectionality in Absolute Geometry, Canad. Math. Bull., 32, 54-63 (1989) · Zbl 0635.51010 |
[7] | Wonnenburger, M. J., Transformations Which Are Products of Two Involutions, J. Math. Mech., 16, 327-338 (1966) · Zbl 0168.03403 |
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