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 Author Reese, Randall Dean Title Topics Pertaining to the Group Matrix: k-Characters and Random Walks URL https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=6569&context=etd Publication Date 2015 Degree MS Degree Level masters University/Publisher Brigham Young University Abstract Linear characters of finite groups can be extended to take k operands. The basics of such a k-fold extension are detailed. We then examine a proposition by Johnson and Sehgal pertaining to these k-characters and disprove its converse. Probabilistic models can be applied to random walks on the Cayley groups of finite order. We examine random walks on dihedral groups which converge after a finite number of steps to the random walk induced by the uniform distribution. We present both sufficient and necessary conditions for such convergence and analyze aspects of algebraic geometry related to this subject. Subjects/Keywords k-characters; group determinant; random walks; branched covering; Mathematics Language en Rights License: http://lib.byu.edu/about/copyright/ Country of Publication us Format application:pdf Record ID oai:scholarsarchive.byu.edu:etd-6569 Repository byu Date Retrieved 2019-12-27 Date Indexed 2019-12-30

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…element of GL(C). Definition 1.6. Another important representation is the regular representation. The regular representation is given by the natural action of G on the group algebra CG. Explicitly, G = {g1 , g2 , . . . , gn } (…

…the proof of Table 1.1: Regular Character for G π g = id 0 id n which employs module theory and will not be discussed herein). Theorem 1.9. For a finite group G, let Irr(G) = {χ1 , χ2 , . . . , χk }. The regular character π…

…characters of a group G is equal to the number of conjugacy classes of G. That is, |Irr(G)| = k, where k is the number of conjugacy classes of G. 3 Characters for arbitrary finite groups were first defined by Frobenius in his 1896 work � Uber…

…Gruppencharaktere  (as cited in ) for use in factoring the group determinant. Chapter 2 will further explicate this topic. Character values for different characters of G can be summarized in a character table. Definition 1.12…

…x5D; The character table of a group G is an invertible matrix. Example. A group of relative importance in this thesis is the dihedral group of order 10, denoted herein as D10 = r, s | r5 = s2 = id, s−1 rs = r−1 . D10 has four conjugacy classes: {…

group determinant provided a major impetus for the development of character theory. We begin this chapter with a collection of definitions and theorems relevant to this pursuit. 2.1 The Group Matrix Definition 2.1.  Let G = {g1 , g2…

…gn } be a finite group of order n. We define the −1 , where the ξg are indeterminates in the ring group matrix as the n × n matrix ξgi gj k C[ξg1 , ξg2 , . . . , ξgn ] corresponding to the gk ∈ G, and the rows and columns of the matrix…

…are indexed by the elements of G. We denote the group matrix by XG . Example.  Let G = S3 . We label the elements of S3 : id = g1 , (123) = g2 , (132) = g3 , (12) = g4 , (13) = g5 , (23…