Minimal generating pairs for permutation groups

Thesis

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_n 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 s_n.

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.

Authors: Conder, M; Conder, Marston Donald Edward

• Publication date: 1980
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Group theory