Bireflectionality of orthogonal and symplectic groups

Ellers, Erich W.; Nolte, Wolfgang

doi:10.1007/bf01899190

Cited by 26 publications

(30 citation statements)

References 5 publications

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“…The following theorem was proved by Wonenburger [11] in odd characteristic, Gow [5] and Ellers and Nolte [3] in even characteristic, and by Djoković [2] in all characteristics.…”

Section: Finding the Involutionsmentioning

confidence: 93%

“…The proof of [3,Lemma 4] shows that N is an orthogonal involution that conjugates S to its inverse. We will show that rankðI þ NÞ is even.…”

Section: Proof (A)mentioning

confidence: 99%

“…The alternating group A n is strongly real if and only if n A f5; 6; 10; 14g (see [1]), and it has been shown in [3], [5], [6] that PSp 2n ðqÞ is strongly real if q 2 3 ðmod 4Þ. Knü ppel and Thomsen [7] have determined which of the orthogonal groups in odd characteristic are strongly real, and Galt [4] has independently obtained the same results for some of the orthogonal groups.…”

Section: Introductionmentioning

confidence: 99%

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Strongly real elements of orthogonal groups in even characteristic

Rämö

2011

Journal of Group Theory

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“…The following theorem was proved by Wonenburger [11] in odd characteristic, Gow [5] and Ellers and Nolte [3] in even characteristic, and by Djoković [2] in all characteristics.…”

Section: Finding the Involutionsmentioning

confidence: 93%

“…The proof of [3,Lemma 4] shows that N is an orthogonal involution that conjugates S to its inverse. We will show that rankðI þ NÞ is even.…”

Section: Proof (A)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Strongly real elements of orthogonal groups in even characteristic

Rämö

2011

Journal of Group Theory

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“…We observe that v, w are linearly independent (otherwise F (π 1 ) = F (π 2 ), a contradiction). Using [6, Theorem W] or [2] or [3], we may assume that B(π) = B(π 1 ) + B(π 2 ). Then and B(ρπ) = B(ι 1 ) + B(ι 2 ).…”

Section: Let π Be An Element Inmentioning

confidence: 99%

The special orthogonal group is trireflectional

Ellers

Villa

2004

Archiv der Mathematik

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Let K be a field of even characteristic, V a finite-dimensional vector space over K, and SO(V ) the special orthogonal group. Then SO(V ) is trireflectional, provided dim V > 2 and SO(V ) = O + (4, 2). Problem statement.Let V be a finite-dimensional vector space over a field K. A mapping q : V −→ K is called a quadratic form if q(αx) = α 2 q(x) for all α ∈ K and all x ∈ V and if there is a symmetric bilinear form b q : V × V −→ K such that b q (x, y) := q(x+y)−q(x)−q(y) for x, y ∈ V . We call the bilinear form b q nondegenerate or nonsingular if b q (x, y) = 0 for all y ∈ V implies x = 0. The orthogonal group of q is defined by

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“…Ambrosiewicz [1] exhibits orthogonal 2 Â 2 and 3 Â 3 matrices, with entries in a field of characteristic 2, which are not products of two involutory orthogonal matrices. On the other hand, Gow [7] as well as Ellers and Nolte [6] and Ellers, Frank, and Nolte [4] showed that every orthogonal transformation on a vector space with nondegenerate quadratic form over a field of characteristic 2 is a product of two involutions. A closer look reveals, the latter authors include the identity in the set of involutions.…”

mentioning

confidence: 99%

Bireflectionality of orthogonal and symplectic groups of characteristic 2

Ellers

1999

Archiv der Mathematik

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Let V be a finite dimensional vector space over a field K of characteristic 2. Let OV be the orthogonal group defined by a nondegenerate quadratic form. Then every element in OV is a product of two elements of order 2, unless all nonsingular subspaces of V are at most 2-dimensional. If V is a nonsingular symplectic space, then every element in the symplectic group Sp V is a product of two elements of order 2, except if dim V 2 and jKj 2X 1. Introduction. Ambrosiewicz [1] exhibits orthogonal 2 Â 2 and 3 Â 3 matrices, with entries in a field of characteristic 2, which are not products of two involutory orthogonal matrices. On the other hand, Gow [7] as well as Ellers and Nolte [6] and Ellers, Frank, and Nolte [4] showed that every orthogonal transformation on a vector space with nondegenerate quadratic form over a field of characteristic 2 is a product of two involutions. A closer look reveals, the latter authors include the identity in the set of involutions. Therefore their result may be reformulated as follows: every transformation is either an element of order 2 or it is a product of two elements of order 2X (The examples mentioned above are easily seen to be matrices of order 2X In view of certain problems, e. g. the question if every element in a finite simple group is a product of two conjugate elements (known as the Thompson conjecture [5]), it is desirable to know if all elements in a group are products of exactly two elements of order 2. This turns out to be true for all orthogonal groups over fields of characteristic 2Y with nondegenerate quadratic forms, except for those on vector spaces V with dim V À dim R 2Y where R denotes the radical of VX A similar result is true for nonsingular symplectic groups.

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Bireflectionality of orthogonal and symplectic groups

Cited by 26 publications

References 5 publications

Strongly real elements of orthogonal groups in even characteristic

Strongly real elements of orthogonal groups in even characteristic

The special orthogonal group is trireflectional

Bireflectionality of orthogonal and symplectic groups of characteristic 2

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