Minimal generating pairs for permutation groups
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 An can be generated by elements x, y which satisfy x² = y³ = (xy)k = 1 and further, if k is even then the same is true of (all but finitely many of) the symmetric groups sn. 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.