Almost-inner automorphisms and almost-conjugate subgroups

Date

1997-08

Journal Title

Journal ISSN

Volume Title

Publisher

Texas Tech University

Abstract

In 1911, Burnside published a paper [3] in the Proceedings of the London Mathematical Society settling a question he had raised while wTiting his Theory of Groups. He wanted to determine if there exists some outer automorphism of a group that maps every element into a conjugate element, something that is more commonly called an almost-inner automorphism. His 1911 paper answered that question in the affirmative. However, that question was considered unresolved from that time until the publication of a result by Wall [14]. He found a different group than Burnside, one that had order 32, while Burnside's groups had order of at least 729.

We have four main objectives in this paper. First, we wish to bring Burnside's groups into modern terminology. Second, we want to give a clearer discussion of the relationship between almost-inner automorphisms and almost-conjugate subgroups, similar to the relationship between inner automorphisms and conjugate subgroups. Third, we want to apply Burnside's groups to generate examples of number fields with the same ("-function. Finally, we shall give a new lower bound for the number of Riemann surfaces that are isospectral, but not isometric.

In Chapter II, we will provide basic definitions, along with some examples of almost-inner automorphisms and almost-conjugate subgroups of an algebraic number field. Chapter III will give an analysis of Burnside's work, and extend it to a general result for nonsingular groups. In Chapter IV we will show how graphs can be used to construct almost-conjugate subgroups. In Chapter V, we use the results from Chapter IV to generate nonisomorphic Number Fields with the same C-function. Finally, in Chapter VI, we will conclude by constructing a new lower bound for the number of isospectral, nonisometric Riemann surfaces.

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Keywords

Isomorphisms (Mathematics), Automorphisms, Betweenness relations (Mathematics), Finite groups

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