An Example of a Non-Closed Connected Subgroup of the Two-Dimensional Vector Space

1937 ◽  
Vol 38 (4) ◽  
pp. 920 ◽  
Author(s):  
E. Livenson
Keyword(s):  
Vector Space ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Connected Subgroup ◽  
Closed Connected Subgroup

scholarly journals THREE-DIMENSIONAL ISOLATED QUOTIENT SINGULARITIES IN EVEN CHARACTERISTIC

2017 ◽  
Vol 60 (2) ◽  
pp. 435-445
Author(s):  
VLADIMIR SHCHIGOLEV ◽  
DMITRY STEPANOV
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Three Dimensional ◽  
Finite Subgroup ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Quotient Singularities ◽  
Characteristic 2

AbstractThis paper is a complement to the work of the second author on modular quotient singularities in odd characteristic. Here, we prove that if V is a three-dimensional vector space over a field of characteristic 2 and G < GL(V) is a finite subgroup generated by pseudoreflections and possessing a two-dimensional invariant subspace W such that the restriction of G to W is isomorphic to the group SL2(𝔽2n), then the quotient V/G is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities that are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.

A CHARACTERISATION OF EUCLIDEAN NORMED PLANES VIA BISECTORS

2018 ◽  
Vol 99 (1) ◽  
pp. 121-129 ◽  
Author(s):  
JAVIER CABELLO SÁNCHEZ ◽  
ADRIÁN GORDILLO-MERINO
Keyword(s):  
Vector Space ◽  
Unit Sphere ◽  
Euclidean Norm ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Strictly Convex ◽  
Normed Planes

Our main result states that whenever we have a non-Euclidean norm $\Vert \cdot \Vert$ on a two-dimensional vector space $X$, there exists some $x\neq 0$ such that for every $\unicode[STIX]{x1D706}\neq 1$, $\unicode[STIX]{x1D706}>0$, there exist $y,z\in X$ satisfying $\Vert y\Vert =\unicode[STIX]{x1D706}\Vert x\Vert$, $z\neq 0$ and $z$ belongs to the bisectors $B(-x,x)$ and $B(-y,y)$. We also give several results about the geometry of the unit sphere of strictly convex planes.

The stabilizer of two-dimensional vector space of 27-dimensional module of type E6 over a field of characteristic two

2020 ◽  
pp. 2150151
Author(s):  
Yousuf Alkhezi ◽  
Mshhour Bani-Ata
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Borel Subgroup ◽  
Unipotent Radical ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Type Formula ◽  
Characteristic Two ◽  
Dimensional Module

The purpose of this paper is to use the notion of [Formula: see text]-sets (cocliques) introduced by the second author in [S. Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type [Formula: see text] for fields of characteristic two, Beitrag Zur Algebra und Geometry 58 (2017) 529–534.] and using Levi components and unipotent radical subgroups of [Formula: see text] to give an elementary and self-contained construction of the stabilizer of two dimensional vector space of 27-dimensional module of type [Formula: see text] over a field of characteristic two. This stabilizer is in fact the maximal parabolic subgroup [Formula: see text] of [Formula: see text] or a Borel subgroup. This construction is elementary on the account that we use not more than little naive linear algebra notions. For more information one can see [M. E. Aschbacher, The 27-dimensional module for [Formula: see text], 1, Invent. Math. 89 (1987) 159–195; M. E. Aschbacher, The 27-dimensional module for [Formula: see text], II, J. London Math. Soc. 37 (1988) 275–293; M. Bani-Ata, On Lie algebras of type [Formula: see text] and [Formula: see text] over finite fields of characteristic two, Preprint; B. Cooperstein, Subgroups of the group [Formula: see text] which are generated by root-subgroups, J. Algebra 46 (1977) 355–388.].

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

scholarly journals Identities Generalizing the Theorems of Pappus and Desargues

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1382
Author(s):  
Roger D. Maddux
Keyword(s):  
Projective Plane ◽  
Vector Space ◽  
Invariant Theory ◽  
Lattice Theory ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

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