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Involutions and commutators in orthogonal groups

Part of: Algebraic groups Forms and linear algebraic groups

Published online by Cambridge University Press:  09 April 2009

Frieder Knüppel  and
Gerd Thomsen
Frieder Knüppel
Affiliation:
Mathematisches Seminar, Ludewig-Meyn-Straβe, 4 D-24098 Kiel, Germany
Gerd Thomsen
Affiliation:
Mathematisches Seminar, Ludewig-Meyn-Straβe, 4 D-24098 Kiel, Germany
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Abstract

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Suppose we are given a regular symmetric bilinear from on a finite-dimensional vector space V over a commutative field K of characteristic ≠ 2. We want to write given elements of the commutator subgroup ω(V) (of the orthogonal group O(V)) and also of the kernel of the spinorial norm ker(Θ) as (short) products of involutions and as products of commutators

MSC classification

Secondary: 11E57: Classical groups 14L35: Classical groups (geometric aspects)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 1 , August 1998 , pp. 1 - 36
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bünger, F., Knüppel, F. and Nielsen, K., ‘Products of symmetries in unitary groups’, Linear Algebra Appl. 260 (1997), 9–42.CrossRefGoogle Scholar
[2]Hahn, A. J., ‘The elements of the orthogonal group ωn (v) as products of commutators of symmetries’, Manuscript, Notre Dame, September 1994.Google Scholar
[3]Hahn, J. and 'Meara, O. T. O, The classical groups and K-theory (Springer, Berlin, 1989).CrossRefGoogle Scholar
[4]Huppert, B., ‘Isometrien von Vektorräumen I’, Arch. Math. 35 (1980), 164–176.CrossRefGoogle Scholar
[5]Huppert, B., ‘Isometrien von Vektorräumen II’, Math. Z. 175 (1980), 5–20.CrossRefGoogle Scholar
[6]Huppert, B., Angewandte lineare Algebra (de Gruyter, Berlin, 1990).CrossRefGoogle Scholar
[7]Ito, N., ‘A theorem on the alternating group An (n ≥ 5)’, Math. Japon. 2 (1951), 59–60.Google Scholar
[8]Knüppel, F., ‘Products of involutions in orthogonal groups’, Ann. Discrete Math. 37 (1988), 231–248.CrossRefGoogle Scholar
[9]Knüppel, F., ‘Products of simple isometries of given conjugacy types’, Forum Math. 5 (1993), 441–458.CrossRefGoogle Scholar
[10]Knüppel, F. and Nielsen, K., ‘On products of two involutions in the orthogonal group of a vector space’, Linear Algebra Appl. 94 (1987), 209–216.CrossRefGoogle Scholar
[11]Knüppel, F., ‘SL(V) is 4-reflectional’, Geom. Dedicata 38 (1991), 301–308.CrossRefGoogle Scholar
[12]Lorenz, F., Quadratische Formen über Körpern, Lecture Notes in Math. 130 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[13]Nielsen, K., ‘Conjugacy classes and commutators in sympletic groups’, to appear.Google Scholar
[14]Ore, O., ‘Some remarks on commutators’, Proc. Amer. Math. Soc. 2 (1951), 307–314.CrossRefGoogle Scholar
[15]Quin, J. M., ‘On commutators in orthogonal groups’, Chinese J. Math. 5 (1965), 437–449.Google Scholar
[16]Scherk, P., ‘On the decomposition of orthogonalities into symmetries’, Proc. Amer. Math. Soc. 1 (1950), 481–491.CrossRefGoogle Scholar
[17]Thompson, R. C., ‘Commutators in the special and general linear groups’, Trans. Amer. Math. Soc. 1 (1950), 481–491.Google Scholar
[18]Thomsen, G., Läangenprobleme in klassischen Gruppen. Teil I: Involutionen mit zwei-dimensionalem Negativraum als Erzeugende der speziellen linearen Gruppe. Teil II: Die Kommutator Gruppe der orthogonalen Gruppen und der Kern der Spinornorm (Dissertation, Kiel, 1991).Google Scholar
[19]Tôyama, H., ‘On commutators of matrices’, Kodai Math. Sem. Rep. 5–6 (1949), 1–2.Google Scholar
[20]Wall, G. E., ‘On the conjugacy classes in the unitary, symplectic and orthogonal groups’, J. Austral. Math. Soc. 3 (1963), 1–62.CrossRefGoogle Scholar
[21]Wonenburger, M. J., ‘Transformations which are products of two involutions’, J. Math. Mech. 16 (1966), 327–338.Google Scholar
[22]Zassenhaus, H., ‘On the spinor norm’, Arch. Math. 13 (1962), 434–451.CrossRefGoogle Scholar