In this thesis we consider two-element generation of certain permutation groups. Interest is focussed mainly on the finite alternating and symmetric groups. Specifically, we prove that if k is any integer greater than six, then all but finitely many of the alternating groups A<sub>n</sub> can be generated by elements x, y which satisfy x² = y³ = (xy)<sup>k</sup> = 1 and further, if k is even then the same is true of (all but finitely many of) the symmetric groups s<sub>n</sub>. The case k = 7 is of particular importance. Any finite group which can be generated by elements x, y satisfying x² = y³ = (xy)⁷ = 1 is called a Hurwitzgroup, and gives rise to a compact Riemann surface of which it is a maximal automorphism group. The bulk of the thesis is devoted to showing that all but 64 of the alternating groups are Hurwitz. Also we give a classification of all Hurwitz groups of order less than one million. An appendix deals with two-element generation of the group associated with the Hungarian 'magic' colour-cube.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:253910 |
| Date | January 1980 |
| Creators | Conder, Marston D. E. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:75cd51df-8c16-4c00-85c2-32d85905164c |
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