scholarly journals Quotient complete intersections of affine spaces by finite linear groups

1985 ◽  
Vol 98 ◽  
pp. 1-36 ◽  
Author(s):  
Haruhisa Nakajima
Keyword(s):  
Normal Subgroup ◽  
Affine Space ◽  
Finite Subgroup ◽  
Complete Intersections ◽  
Linear Groups ◽  
Coordinate Ring ◽  
Affine Spaces ◽  
Quotient Variety ◽  
Finite Linear ◽  
Complex Number Field

Let G be a finite subgroup of GLn(C) acting naturally on an affine space Cn of dimension n over the complex number field C and denote by Cn/G the quotient variety of Cn under this action of G. The purpose of this paper is to determine G completely such that Cn/G is a complete intersection (abbrev. CI.) i.e. its coordinate ring is a C.I. when n > 10. Our main result is (5.1). Since the subgroup N generated by all pseudo-reflections in G is a normal subgroup of G and Cn/G is obtained as the quotient variety of without loss of generality, we may assume that G is a subgroup of SLn(C) (cf. [6, 16, 24, 25]).

On Complete Intersections Over an Algebraically Non-Closed Field

1986 ◽  
Vol 29 (2) ◽  
pp. 140-145 ◽  
Author(s):  
Maria Grazia Marinari ◽  
Mario Raimondo
Keyword(s):  
Complete Intersection ◽  
Affine Space ◽  
Complete Intersections ◽  
Closed Field ◽  
Affine Variety ◽  
Affine Spaces ◽  
Real Curve

AbstractWe give a criterion in order that an affine variety defined over any field has a complete intersection (ci.) embedding into some affine space. Moreover we give an example of a smooth real curve C all of whose embeddings into affine spaces are c.i.; nevertheless it has an embedding into ℝ3 which cannot be realized as a c.i. by polynomials.

scholarly journals SOME INFINITE PERMUTATION GROUPS AND RELATED FINITE LINEAR GROUPS

2016 ◽  
Vol 102 (1) ◽  
pp. 136-149 ◽  
Author(s):  
PETER M. NEUMANN ◽  
CHERYL E. PRAEGER ◽  
SIMON M. SMITH
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Finite Rank ◽  
Minimal Condition ◽  
Permutation Groups ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Second Variation ◽  
Formula Group ◽  
Finite Linear

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

Finite Linear Groups of Degree Six

1971 ◽  
Vol 23 (5) ◽  
pp. 771-790 ◽  
Author(s):  
J. H. Lindsey
Keyword(s):  
Tensor Product ◽  
Finite Groups ◽  
Existence And Uniqueness ◽  
Projective Representation ◽  
Linear Groups ◽  
3 Dimensional ◽  
Dimensional Group ◽  
Complex Linear ◽  
Finite Linear ◽  
Complex Number Field

In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3, 4)|, and |PSL4(3)|. By [19], X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representation of a simple group, possibly extended by some automorphisms. The tensor product case is discussed in section 10. Otherwise, we assume that G/Z(G) is simple. We discuss which automorphisms of G/Z(G) extend the representation X (that is, lift to the central extension G and fix the character corresponding to X) just after we find X(G).

scholarly journals Dynamical degrees of affine-triangular automorphisms of affine spaces

2021 ◽  
pp. 1-42
Author(s):  
JÉRÉMY BLANC ◽  
IMMANUEL VAN SANTEN
Keyword(s):  
Affine Space ◽  
Affine Spaces ◽  
Dynamical Degree ◽  
Perron Number ◽  
Dynamical Degrees ◽  
Affine Automorphism

Abstract We study the possible dynamical degrees of automorphisms of the affine space $\mathbb {A}^n$ . In dimension $n=3$ , we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb {A}^n$ for some n, and we give the best possible n for quadratic integers, which is either $3$ or $4$ .

The subnormal structure of general linear groups

1986 ◽  
Vol 99 (3) ◽  
pp. 425-431 ◽  
Author(s):  
L. N. Vaserstein
Keyword(s):  
Normal Subgroup ◽  
Natural Number ◽  
Associative Ring ◽  
Inverse Image ◽  
General Linear ◽  
Linear Groups ◽  
Canonical Homomorphism ◽  
General Linear Groups ◽  
Subnormal Structure ◽  
Image Position

Let A be an associative ring with 1. For any natural number n, let GLnA denote the group of invertible n by n matrices over A, and let EnA be the subgroup generated by all elementary matrices ai, j, where aεA and 1 ≤ i ≡ j ≤ n. For any (two-sided) ideal B of A, let GLnB be the kernel of the canonical homomorphism GLnA→GLn(A/B) and Gn(A, B) the inverse image of the centre of GLn(A/B) (when n > 1, the centre consists of scalar matrices over the centre of the ring A/B). Let EnB denote the subgroup of GLnB generated by its elementary matrices, and let En(A, B) be the normal subgroup of EnA generated by EnB (when n > 2, the group GLn(A, B) is generated by matrices of the form ai, jbi, j(−a)i, j with aA, b in B, i ≡ j, see [7]). In particuler,is the centre of GLnA

scholarly journals Prime Power Representations of Finite Linear Groups II

1957 ◽  
Vol 9 ◽  
pp. 347-351 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Lie Groups ◽  
Structural Properties ◽  
Lie Group ◽  
Prime Power ◽  
Linear Groups ◽  
Finite Linear

The aim of this paper is two-fold: first, to extend the results of (4) to the exceptional finite Lie groups recently discovered by Chevalley (1), and, secondly, to give a construction which works simultaneously for the groups An, Bn, Cn, Dn, En, F4 and G2 (in the usual Lie group notation), and which depends only on intrinsic structural properties of these groups.

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