scholarly journals Finite linear groups containing an irreducible solvable normal subgroup

1970 ◽  
Vol 25 (3) ◽  
pp. 716
Author(s):  
David L. Winter
Keyword(s):  
Normal Subgroup ◽  
Linear Groups ◽  
Finite Linear

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]).

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.

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.

scholarly journals Commutator Length of Finitely Generated Linear Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-5
Author(s):  
Mahboubeh Alizadeh Sanati
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Quadratic Polynomial ◽  
Linear Groups ◽  
Number Of Generators ◽  
Derived Subgroup ◽  
Finite Linear Group ◽  
Minimum Number ◽  
Commutator Length ◽  
Finite Linear

The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

Prime Power Representations Of Finite Linear Groups

1956 ◽  
Vol 8 ◽  
pp. 580-591 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Finite Groups ◽  
Symplectic Group ◽  
Prime Power ◽  
Projective Group ◽  
Linear Groups ◽  
Orthogonal Groups ◽  
Group 2 ◽  
Finite Linear ◽  
Two Parameter ◽  
Group 1

1. Introduction. There are five well-known, two-parameter families of simple finite groups: the unimodular projective group, the symplectic group,1 the unitary group,2 and the first and second orthogonal groups, each group acting on a vector space of a finite number of elements (2; 3).

On finite linear groups

1968 ◽  
Vol 106 (4) ◽  
pp. 245-247 ◽  
Author(s):  
D. L. Winter
Keyword(s):  
Linear Groups ◽  
Finite Linear
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