Centralizers of irreducible subgroups in the projective linear group

2018 ◽  
Vol 21 (5) ◽  
pp. 789-816
Author(s):  
Clément Guérin
Keyword(s):  
Linear Group ◽  
Finite Graph ◽  
Conjugacy Classes ◽  
Combinatorial Properties ◽  
Isomorphism Classes ◽  
Character Varieties ◽  
Projective Linear Group

Abstract In this paper, we classify conjugacy classes of centralizers of irreducible subgroups in {\mathrm{PGL}(n,\mathbb{C})} using alternate modules. When n is square-free, we prove that these conjugacy classes are classified by their isomorphism classes. More generally, we define a finite graph related to this classification whose combinatorial properties are expected to help describe the stratification of the bad locus of some character varieties.

Flag-Transitive 3- (v, k,3) Designs and PSL(2,q) Groups

2021 ◽  
Vol 28 (01) ◽  
pp. 33-38
Author(s):  
Shaojun Dai ◽  
Shangzhao Li
Keyword(s):  
Linear Group ◽  
Automorphism Groups ◽  
Two Dimensional ◽  
Text Design ◽  
Projective Linear Group

This article is a contribution to the study of the automorphism groups of 3-[Formula: see text] designs. Let [Formula: see text] be a non-trivial 3-[Formula: see text] design. If a two-dimensional projective linear group [Formula: see text] acts flag-transitively on [Formula: see text], then [Formula: see text] is a 3-[Formula: see text] or 3-[Formula: see text] design.

scholarly journals The Energy of Conjugacy Classes Graphs of Some Order of Alternating Groups

2020 ◽  
Vol 7 (4) ◽  
pp. 62-71
Author(s):  
Zuzan Naaman Hassan ◽  
Nihad Titan Sarhan
Keyword(s):  
Conjugacy Class ◽  
Adjacency Matrix ◽  
Finite Graph ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Alternating Groups ◽  
Eigen Values ◽  
Finite Set ◽  
Class Graph ◽  
Square Matrix

The energy of a graph , is the sum of all absolute values of the eigen values of the adjacency matrix which is indicated by . An adjacency matrix is a square matrix used to represent of finite graph where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. The group of even permutations of a finite set is known as an alternating group  . The conjugacy class graph is a graph whose vertices are non-central conjugacy classes of a group , where two vertices are connected if their cardinalities are not coprime. In this paper, the conjugacy class of alternating group  of some order for   and their energy are computed. The Maple2019 software and Groups, Algorithms, and Programming (GAP) are assisted for computations.

Conjugacy classes of torsion in GL_n(Z)

2015 ◽  
Vol 30 ◽  
Author(s):  
Qingjie Yang
Keyword(s):  
Linear Group ◽  
Equivalence Relation ◽  
General Linear Group ◽  
Conjugacy Classes ◽  
Complete List ◽  
General Linear ◽  
Triangular Matrices ◽  
Torsion Element ◽  
Ring Of Integers

The problem of integral similarity of block-triangular matrices over the ring of integers is connected to that of finding representatives of the classes of an equivalence relation on general integer matrices. A complete list of representatives of conjugacy classes of torsion in the 4 × 4 general linear group over ring of integers is given. There are 45 distinct such classes and each torsion element has order of 1, 2, 3, 4, 5, 6, 8, 10 or 12.

scholarly journals Weight Parameterization of Simple Modules for p-Solvable Groups

2013 ◽  
Vol 20 (01) ◽  
pp. 1-46
Author(s):  
Lluis Puig
Keyword(s):  
Solvable Groups ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Characteristic P ◽  
Isomorphism Classes ◽  
Outer Automorphisms ◽  
Odd Order ◽  
The One ◽  
A Finite Group ◽  
The Relationship

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of isomorphism classes of simple kG-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.

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