scholarly journals Linear groups analogous to permutation groups

1963 ◽  
Vol 3 (2) ◽  
pp. 180-184 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Linear Group ◽  
Singular Matrices ◽  
A Finite Group ◽  
Complex Coefficients ◽  
Finite Linear

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), I shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n.

scholarly journals The minimal degree of a faithful quasi-permutation representation of an abelian group

1997 ◽  
Vol 39 (1) ◽  
pp. 51-57 ◽  
Author(s):  
Houshang Behravesh
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Linear Group ◽  
Singular Matrices ◽  
A Finite Group ◽  
Complex Coefficients ◽  
Finite Linear

Let G be a finite linear group of degree n; that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients). We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n -dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [5].

scholarly journals On quasi-permutation representations of finite groups

1994 ◽  
Vol 36 (3) ◽  
pp. 301-308 ◽  
Author(s):  
J. M. Burns ◽  
B. Goldsmith ◽  
B. Hartley ◽  
R. Sandling
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Minimal Degree ◽  
Permutation Representation ◽  
Permutation Groups ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Group Situation ◽  
A Finite Group

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.

Some Results on the Schur Index of a Representation of a Finite Group

1970 ◽  
Vol 22 (3) ◽  
pp. 626-640 ◽  
Author(s):  
Charles Ford
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Complex Field ◽  
Complex Vector Space ◽  
Linear Transformations ◽  
Complex Vector ◽  
Finite Dimensional ◽  
A Finite Group

Let ℭ be a finite group with a representation as an irreducible group of linear transformations on a finite-dimensional complex vector space. Every choice of a basis for the space gives the representing transformations the form of a particular group of matrices. If for some choice of a basis the resulting group of matrices has entries which all lie in a subfield K of the complex field, we say that the representation can be realized in K. It is well known that every representation of ℭ can be realized in some algebraic number field, a finitedimensional extension of the rational field Q.

Reflection Subquotients of Unitary Reflection Groups

1999 ◽  
Vol 51 (6) ◽  
pp. 1175-1193 ◽  
Author(s):  
G. I. Lehrer ◽  
T. A. Springer
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Vector Fields ◽  
Reflection Group ◽  
Complex Vector Space ◽  
Reflection Groups ◽  
Complex Vector ◽  
Invariant Vector ◽  
A Finite Group ◽  
Invariant Vector Fields

AbstractLet G be a finite group generated by (pseudo-) reflections in a complex vector space and let g be any linear transformation which normalises G. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset gG, a subquotient of G which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in G of certain elements of the coset. A criterion is also given in terms of the invariant degrees of G for an integer to be regular for G. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

Linear Symmetry Classes

1976 ◽  
Vol 28 (6) ◽  
pp. 1311-1319 ◽  
Author(s):  
L. J. Cummings ◽  
R. W. Robinson
Keyword(s):  
Vector Space ◽  
Permutation Group ◽  
Cyclic Group ◽  
Symmetry Class ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Linear Character ◽  
Finite Permutation Group ◽  
Symmetry Classes ◽  
Finite Dimensional

A formula is derived for the dimension of a symmetry class of tensors (over a finite dimensional complex vector space) associated with an arbitrary finite permutation group G and a linear character of x of G. This generalizes a result of the first author [3] which solved the problem in case G is a cyclic group.

scholarly journals A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

2020 ◽  
Vol 32 (3) ◽  
pp. 713-721
Author(s):  
Andrea Lucchini ◽  
Mariapia Moscatiello ◽  
Pablo Spiga
Keyword(s):  
Finite Group ◽  
Permutation Group ◽  
Maximal Subgroups ◽  
Transitive Permutation Group ◽  
Classic Result ◽  
A Finite Group

AbstractWe show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by {a\lvert G:H\rvert^{3/2}}. In particular, a transitive permutation group of degree n has at most {an^{3/2}} maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is, the number of maximal subgroups of G containing H is at most {\lvert G:H\rvert-1}.

scholarly journals Groups fixing graphas in switching classes

1982 ◽  
Vol 33 (1) ◽  
pp. 76-85
Author(s):  
Martin W. Liebeck
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Permutation Group ◽  
Abelian Groups ◽  
Finite Set ◽  
A Finite Group ◽  
Switching Class

AbstractA permutation group G on a finite set Ω is always exposable if whenever G stabilises a switching class of graphs on Ω, G fixes a graph in the switching class. Here we consider the problem: given a finite group G, which permutation representations of G are always exposable? We present solutions to the problem for (i) 2-generator abelian groups, (ii) all abelian groups in semiregular representations. (iii) generalised quaternion groups and (iv) some representations of the symmetric group Sn.

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 .

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