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Journal of Pure and Applied Algebra on Subgroups of Orthogonal Groups Journal of Pure and Applied Algebra 214 (2010) 1049–1062 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa On subgroups of orthogonal groups of isotropic quadratic forms Evgenii L. Bashkirov Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan article info a b s t r a c t Article history: Let a field K be an algebraic extension of a subfield k of characteristic not 2, n an integer, Received 1 December 2008 n ≥ 3; Q a non-degenerate isotropic form in n variables over K with coefficients in k. Available online 21 October 2009 We study subgroups of the orthogonal group On.K; Q / that contain the derived subgroup Communicated by R. Parimala Ωn.k; Q / of the group On.k; Q /. ' 2009 Elsevier B.V. All rights reserved. MSC: 20G15 20H25 1. Introduction Let K be a field extension of a field k of characteristic not two, n an integer, n ≥ 2; Q a non-degenerate quadratic form in n indeterminates over k. If L is an arbitrary field extension of k, then we denote by QL the quadratic form which is the scalar extension of Q by L. If the field L is clear from the context, we write simply Q instead of QL. Note that the Witt index of QL (the dimension of a maximal totally isotropic subspace of a vector L-space equipped with QL) may be greater than that of Q . By Ωn.L; QL/ D Ωn.L; Q / we denote the derived subgroup of the orthogonal group On.L; QL/ D On.L; Q / defined by QL. The purpose of the present paper is to describe subgroups of On.K; Q / containing Ωn.k; Q / provided the form Q D Qk is isotropic, i.e., its Witt index is different from zero. The similar problem for the orthogonal groups of quadratic forms whose Witt index is greater than one has been settled by Ya.N. Nuzhin [12] within the framework of his general investigation of Chevalley groups whose rank is greater than one. When n ≥ 2 and Q has non-zero Witt index, O. King [9] has examined groups which lie between the special orthogonal group SOn.k; Q /, that is, the group of all elements in On.k; Q / whose determinant equals to one and the special linear group SLn.k/. It is worthwhile to note that under the condition n D 2 the result from [9] gives in fact the remarkable description of those subgroups of SL2.k/ that contain the subgroup of monomial matrices. Overgroups of Ωn.k; Q / in the general linear group GLn.K/ for n > 4 have been studied by the author in [1,2] provided that the Witt index of Q is greater than one and the field K is an algebraic extension of k. Regarding the groups Ωn.k; Q / of quadratic forms Q whose Witt index is one, the most essential result in this area seem to be obtained by Li Shang Zhi [10] who has described overgroups of Ωn.k; Q / in SLn.k/. The present paper is devoted to proving the following theorem which may be regarded as the further development of the above mentioned results. Theorem 1.1. Let a field K be an algebraic extension of a subfield k of characteristic not two, n an integer, n ≥ 3; Q a non- degenerate isotropic quadratic form in n variables over k. Suppose that k is infinite. Let Ωn.k; Q / ≤ X ≤ On.K; Q /. Then one of the following assertions holds: (1) If n 6D 4, then X contains a normal subgroup Ωn.L; Q /, where L is a subfield of K such that k ⊆ L. (2) If n D 4 and QK has Witt index 1, then X contains a normal subgroup isomorphic to the group ∼ Ω3.L0; Q0/ D PSL2.L0/ D SL2.L0/={±1g; E-mail address: [email protected]. 0022-4049/$ – see front matter ' 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2009.09.010 1050 E.L. Bashkirov / Journal of Pure and Applied Algebra 214 (2010) 1049–1062 where L0 is a subfield of some quadratic field extension of K; L0 ⊇ k; Q0 is a non-degenerate quadratic isotropic form in three variables over L0; in particular, if K is a simple extension of k, then X contains a normal subgroup Ω4.L; Q /, where L is a subfield of K containing k. (3) If n D 4 and Q is a quadratic form of Witt index 2, then X contains a normal subgroup isomorphic either to the group ∼ K Ω3.L0; Q0/ D PSL2.L0/, where L0 is a subfield of the direct sum K ⊕ K of two copies of the field K and Q0 is a non-degenerate isotropic form in three variables over L0 or to the quotient group .SL2.L1/ × SL2.L2//={±1g, where L1 and L2 are subfields of K that contain k. The rather large size of formulation of Theorem 1.1 is caused by the particular structure of the group Ω4. If the constraint n 6D 4 is imposed, the theorem takes a less intricate form. Actually it keeps only Part (1). Note that Theorem 1.1 includes the case of quadratic forms whose Witt index is one. Thereby this theorem generalizes and strengthens those results of [12] which relate to the orthogonal groups. Also Theorem 1.1 combined with the above mentioned results from [1,2,9,10,12] may be regarded as a part of the direction of the group theory that studies the subgroup structure of linear groups by describing intermediate groups which lie between two given linear groups. Further information on this area which is rather actively elaborated together with related references may be found in the surveys [11,15–18]. The starting point of our considerations is a well known fact that the group On.K; Q / contains unipotent elements of special kind, namely so-called short root elements which may be defined as follows. Let E be a finite-dimensional vector space over a field K of characteristic not two. We endow E with a non-degenerate isotropic quadratic form Q . Denote by f the non-degenerate symmetric bilinear form on E × E which is associated with Q and for every x 2 E satisfies the condition Q .x/ D f .x; x/. In other words, 1 f .x; y/ D TQ .x C y/ − Q .x/ − Q .y/U 8x; y 2 E: (1.1) 2 Further, let u; w be linearly independent vectors in E such that Q .u/ D f .u; w/ D 0. For each x 2 E, we set 1 tu;w.x/ D x C uf .x; w/ − wf .x; u/ − u f .x; u/Q .w/: 2 Here scalars are multiplied from the right, that is, we consider E as a right vector space. We prefer to distinguish between left and right vector spaces to avoid confusions because in our discussion left and right vector spaces may occur simultaneously (for instance, while considering dual vector spaces). The transformation tu;w is an automorphism of E and tu;w 2 Ωn.K; Q / [7]. If Q .w/ 6D 0 (resp. Q .w/ D 0), then tu;w is called a short (resp. long) root element of Ωn.K; Q /. If g is a long root element 2 3 of Ωn.K; Q /, then .g − 1/ D 0, while if g is a short root element of Ωn.K; Q /, the condition .g − 1/ D 0 is satisfied though 2 .g − 1/ 6D 0. It is important to note that the group Ωn.K; Q / contains long root elements if and only if the Witt index of Q is greater than one. It is this circumstance that dictates the choice of short root elements as a main tool used in studying the orthogonal groups corresponding to quadratic forms whose Witt index equals one. The significant property of these elements is also that they generate the group Ωn.K; Q /. The method of the proof of Theorem 1.1 given in the present paper is based on the fact that Ωn.K; Q / is a quotient group of the spinor group Spinn.K; Q /, the short root elements being pre- images of quadratic unipotent elements under the canonical epimorphism Spinn.K; Q / ! Ωn.K; Q /. In turn the behavior and properties of quadratic unipotent elements can be rather well studied and understood by methods which have been developed and advanced before. Complying with this scheme, we derive Theorem 1.1 from the following statement. Theorem 1.2. Let K; k; n; Q be as in Theorem 1.1 and let Spinn.k; Q / ≤ Y ≤ Spinn.K; Q /. The following assertions are true: (1) If n 6D 4, then Y contains a normal subgroup Spin .L; Q /, where L is a subfield of K; k ⊆ L. n ∼ (2) If n D 4 and QK has Witt index 1, then Y contains a normal subgroup isomorphic to the group Spin3.L0; Q0/ D SL2.L0/, where L0 and Q0 are as in Part (2) of Theorem 1.1. In particular, if K is a simple extension of k, then Y contains a normal subgroup Spin .L; Q /, where L is a subfield of K containing k. 4 ∼ (3) If n D 4 and QK has Witt index 2, then Y contains a normal subgroup isomorphic either to the group Spin3.L0; Q0/ D SL2.L0/, where L0 and Q0 are as in Part (3) of Theorem 1.1 or to the direct product SL2.L1/ × SL2.L2/, where L1 and L2 are subfields of K that contain k.
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