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Title:  Multiplicityfree permutation representations of the alternating groups 
Author(s):  Balmaceda, Jose Maria P. 
Doctoral Committee Chair(s):  Suzuki, Michio 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  A transitive permutation representation of a finite group G is said to be multiplicityfree if each irreducible constituent of the associated permutation character of G occurs with multiplicity one. If H is a subgroup of G, then G acts naturally on the set of cosets of H. This action is transitive and the associated permutation character is given by the character 1$\rm \sbsp{H}{G}$ of G induced from the trivial character of H. If 1$\rm \sbsp{H}{G}$ is multiplicityfree, then H is said to be a multiplicityfree subgroup of G. In this paper the multiplicityfree subgroups H of the alternating groups A$\sb{\rm n}$, n $>$ 18, are investigated and classified in the main chapter. The key tools involve bounds on the order of H and the analysis of the orbits of H on kelement subsets of the set of n elements on which A$\sb{\rm n}$ acts, k $\leq$ n. The multiplicityfree subgroups of A$\sb{\rm n}$ are shown to have large orders and to be highly transitive. Explicit decompositions of several permutation characters are obtained. A survey of the known results on multiplicityfree permutation representations is included. It is shown that 1$\rm \sbsp{H}{G}$ is multiplicityfree if and only if the algebra of complexvalued functions on G which are constant on the (H,H)double cosets under the convolution product is commutative. The author proves that if G is a group of odd order which admits an involutary automorphism, and H is the subgroup of fixed points, then 1$\rm \sbsp{H}{G}$ is multiplicityfree. 
Issue Date:  1991 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/22828 
Rights Information:  Copyright 1991 Balmaceda, Jose Maria P. 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9136537 
OCLC Identifier:  (UMI)AAI9136537 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois