scholarly journals Abelian 2-subgroups of finite symplectic groups in characteristic 2

1982 ◽  
Vol 33 (3) ◽  
pp. 345-350 ◽  
Author(s):  
M. J. J. Barry ◽  
W. J. Wong
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Symplectic Group ◽  
Preceding Paper ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Characteristic 2 ◽  
Finite Symplectic Groups

AbstractIf Sp(V) is the symplectic group of a vector space V over a finite field of characteristic p, and r is a positive integer, the abelian p-subgroups of largest order in Sp(V) whose fixed subspaces in V have dimension at least r were determined in the preceding paper, in the case p ≠ 2. Here we deal with the case p = 2. Our results also complete earlier work on the orthogonal groups.

scholarly journals Abelian Unipotent Subgroups of Finite Unitary and Symplectic Groups

1982 ◽  
Vol 33 (3) ◽  
pp. 331-344 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Unitary Group ◽  
Symplectic Group ◽  
General Linear ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Abelian Subgroups ◽  
Symplectic Case

AbstractIf G is the unitary group U(V) or the symplectic group Sp(V) of a vector space V over a finite field of characteristic p, and r is a positive integer, we determine the abelian p-subgroups of largest order in G whose fixed subspaces in V have dimension at least r, with the restriction that we assume p ≠ 2 in the symplectic case. In particular, we determine the abelian subgroups of largest order in a Sylow p-subgroup of G. Our results complement earlier work on general linear and orthogonal groups.

scholarly journals Irreducible subgroups of symplectic groups in characteristic 2

2002 ◽  
Vol 73 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Christopher Parker ◽  
Peter Rowley
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Symplectic Group ◽  
Symplectic Groups ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Characteristic 2 ◽  
Zero Vector

AbstractSuppose that V is a finite dimensional vector space over a finite field of characteristic 2, G is the symplectic group on V and a is a non-zero vector of V. Here we classify irreducible subgroups of G containing a certain subgroup of O2(StabG(a)) all of whose non-trivial elements are 2-transvections.

VECTOR INVARIANT IDEALS OF ABELIAN GROUP ALGEBRAS UNDER THE ACTION OF THE SYMPLECTIC GROUPS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350046
Author(s):  
JIZHU NAN ◽  
LINGLI ZENG
Keyword(s):  
Abelian Group ◽  
Finite Field ◽  
Vector Space ◽  
Group Algebra ◽  
Symplectic Group ◽  
Group Algebras ◽  
Symplectic Groups ◽  
Vector Invariant

Let F be a finite field and let Sp 2ν(F) be the symplectic group over F. If Sp 2ν(F) acts on the F-vector space F2ν, then it can induce an action on the vector space F2ν ⊕ F2ν, defined by (x, y)A = (xA, yA), ∀ x, y ∈ F2ν, A ∈ Sp 2ν(F). If K is a field with char K ≠ char F, then Sp 2ν(F) also acts on the group algebra K[F2ν ⊕ F2ν]. In this paper, we determine the structures of Sp 2ν(F)-stable ideals of the group algebra K[F2ν ⊕ F2ν] by augmentation ideals, and describe the relations between the invariant ideals of K[F2ν] and the vector invariant ideals of K[F2ν ⊕ F2ν].

scholarly journals On the dimension of $H^{*}((\mathbb Z_2)^{\times t}, \mathbb Z_2)$ as a module over Steenrod ring

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Positive Integer ◽  
Tensor Product ◽  
Finite Field ◽  
Vector Space ◽  
Linear Group ◽  
Computer Algebra ◽  
General Linear Group ◽  
Polynomial Algebra ◽  
Minimal Set ◽  
Trivial Subgroup

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $(\mathbb P^{\otimes t})^{G_t}$ as a module over Steenrod ring, $\mathscr A_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t, \mathbb Z_2).$ Equivalently, we want to find a basis of the $\mathbb Z_2$-vector space $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each degree $n\geq 0.$ The problem is proved surprisingly difficult and has been not yet known for $t\geq 5.$ In the present paper, we consider the trivial subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators for $(\mathbb P^{\otimes 5})^{G_5}$ in degree $5(2^{1} - 1) + 13.2^{1}$ and for $(\mathbb P^{\otimes 6})^{G_6}$ in "generic" degree $n = 5(2^{d+4}-1) + 47.2^{d+4}$ with a positive integer $d.$ An efficient approach to studying $(\mathbb P^{\otimes 5})^{G_5}$ in this case has been provided. In addition, we introduce an algorithm on the MAGMA computer algebra for the calculation of this space. This study is a continuation of our recent works in \cite{D.P2, D.P4}.

scholarly journals Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
You Gao ◽  
XinZhi Fu
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Vector ◽  
Orthogonal Groups ◽  
Reverse Inclusion ◽  
Dimensional Vector Space ◽  
Geometric Lattices

Let𝔽q(2ν+δ+l)be a(2ν+δ+l)-dimensional vector space over the finite field𝔽q. In this paper we assume that𝔽qis a finite field of odd characteristic, andO2ν+δ+l,  Δ(𝔽q)the singular orthogonal groups of degree2ν+δ+lover𝔽q. Letℳbe any orbit of subspaces underO2ν+δ+l,  Δ(𝔽q). Denote byℒthe set of subspaces which are intersections of subspaces inℳ, where we make the convention that the intersection of an empty set of subspaces of𝔽q(2ν+δ+l)is assumed to be𝔽q(2ν+δ+l). By orderingℒby ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these latticesℒare geometric lattices.

Density of Polynomial Maps

2010 ◽  
Vol 53 (2) ◽  
pp. 223-229 ◽  
Author(s):  
Chen-Lian Chuang ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Prime Ring ◽  
Division Ring ◽  
Polynomial Identity ◽  
Matrix Ring ◽  
Polynomial Maps ◽  
Finite Matrix ◽  
Nonzero Polynomial

AbstractLet R be a dense subring of End(DV), where V is a left vector space over a division ring D. If dimDV = ∞, then the range of any nonzero polynomial ƒ (X1, … , Xm) on R is dense in End(DV). As an application, let R be a prime ring without nonzero nil one-sided ideals and 0 ≠ a ∈ R. If a f (x1, … , xm)n(xi) = 0 for all x1, … , xm ∈ R, where n(xi ) is a positive integer depending on x1, … , xm, then ƒ (X1, … , Xm) is a polynomial identity of R unless R is a finite matrix ring over a finite field.

Defining Relations in Orthogonal Groups of Characteristic Two

1979 ◽  
Vol 31 (6) ◽  
pp. 1217-1246 ◽  
Author(s):  
Georg Güunther ◽  
Wolfgang Nolte
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Orthogonal Groups ◽  
Prime Field ◽  
Defining Relations ◽  
Characteristic 2 ◽  
Characteristic Two

Introduction. It is well-known that a group is uniquely determined by a system of generators, and a set of defining relations on those generators. Clearly it is of interest to find relations that are as simple as possible. In this paper, this question is dealt with for certain orthogonal groups of characteristic 2, which are generated by involutions.Let V be a vector space over a field K of characteristic 2 (we always exclude the prime field K = G F (2)). Let Q be a quadratic form over V, and let S be the set of orthogonal transformations of (V, Q) whose path is 1-dimensional and not contained in the radical of V. Letting O* be the group generated by S, we shall show that every relation among generators in S is a consequence of relations of length 2, 3, or 4.

scholarly journals Involutions and commutators in orthogonal groups

1998 ◽  
Vol 65 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Frieder Knüppel ◽  
Gerd Thomsen
Keyword(s):  
Vector Space ◽  
Orthogonal Group ◽  
Commutator Subgroup ◽  
Dimensional Vector ◽  
Orthogonal Groups ◽  
Commutative Field ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Characteristic 2

AbstractSuppose 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

Subgroups of direct products of groups invariant under the action of permutations on factors

2020 ◽  
Vol 30 (4) ◽  
pp. 243-255
Author(s):  
Dmitry A. Burov
Keyword(s):  
Finite Field ◽  
Finite Number ◽  
Vector Space ◽  
Direct Product ◽  
Subdirect Product ◽  
Additive Group ◽  
Direct Products ◽  
Products Of Groups ◽  
Characteristic 2 ◽  
Additional Constraints

AbstractWe study subgroups of the direct product of two groups invariant under the action of permutations on factors. An invariance criterion for the subdirect product of two groups under the action of permutations on factors is put forward. Under certain additional constraints on permutations, we describe the subgroups of the direct product of a finite number of groups that are invariant under the action of permutations on factors. We describe the subgroups of the additive group of vector space over a finite field of characteristic 2 which are invariant under the coordinatewise action of inversion permutation of nonzero elements of the field.

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