The process of coset enumeration has become a significant factor in
group theoretical investigations since the advent of modern computing
power, but in some respects the process is still not well understood.
This thesis investigates some features of coset enumeration, working
mainly with the group F(2,7).
Chapter 1 describes the characteristics of coset enumeration and
algorithms used for it. A worked example of the method is provided.
Chapter 2 discusses some features which would be desirable in computer
programs for use in investigating the coset enumeration process itself,
and reviews the Havas/Alford program which to date best meets the
requirements.
Chapter 3 deals with the use of coset ammeration in proofs, either in
its own right or as a basis for other workings. An example of one
attempt to obtain a proof by coset enumeration is given.
Chapter 4 reviews techniques designed to reduce the length of coset
enumerations and proposes the 'equality list' technique as a way to
reduce enumeration length for some groups. Extra insights obtainable
using the equality list method are also discussed.
Chapter 5 summarises the factors by which the success of different
coset enumerations can be compared and proposes an algorithm for making
systematic comparisons among enumerations.
Chapter 6 reports five coset enumerations, obtained manually by three
main methods on the group F(2,7). All these enumerations were shorter
than is so far obtainable by machine and one is shorter than other
known hand enumerations. The enumerations were compared by applying
the process developed in Chapter 5.
Chapter 7 presents a shorter proof of the cyclicity of the group F(2,7)
than was hitherto available. The proof derives from the workings for
one of the coset enumerations described in Chapter 6.
There are eight appendices and an annotated bibliography. The
appendices contain, inter alia, edited correspondence between
well-known coset-enumerators, a guide to the Havas/Alford program,
further details on the equality list method and listings of various
enumerations.
| Identifer | oai:union.ndltd.org:ADTP/218647 |
| Date | January 1989 |
| Creators | Edeson, Margaret, n/a |
| Publisher | University of Canberra. Information Sciences & Engineering |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | ), Copyright Margaret Edeson |
Page generated in 0.002 seconds