Classification of regular maps of prime characteristic revisited: Avoiding the Gorenstein-Walter theorem

Breda, Nedela and Širáň (2005) classified the regular maps on surfaces of Euler characteristic $-p$ for every prime $p$. This classification relies on three key theorems, each proved using the highly non-trivial characterisation of finite groups with dihedral Sylow 2-subgroups, due to D. Gorenstein and J.H. Walter (1965). Here we give new proofs of those three facts (and hence the entire classification) using somewhat more elementary group theory, using without referring to the Gorenstein-Walter theorem

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 satisfyx² = y³ = (xy)k = 1and 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 satisfyingx² = y³ = (xy)⁷ = 1is 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.</p

Vertex-transitive non-Cayley graphs with arbitrarily large vertexstabilizer,

Abstract. A construction is given for an infinite family { n } of finite vertex-transitive non-Cayley graphs of fixed valency with the property that the order of the vertex-stabilizer in the smallest vertex-transitive group of automorphisms of n is a strictly increasing function of n. For each n the graph is 4-valent and arc-transitive, with automorphism group a symmetric group of large prime degree p &gt; 2 2 n +2 . The construction uses Sierpinski&apos;s gasket to produce generating permutations for the vertex-stabilizer (a large 2-group)

Regular maps and hypermaps of Euler characteristic -1 to -200

This paper describes the determination of all orientably-regular maps and hypermaps of genus 2 to 101, and all non-orientable regular maps and hypermaps of genus 3 to 202. It extends the lists obtained by Conder and Dobcsányi (2001) of all such maps of Euler characteristic −1 to −28, and corrects errors made in those lists for the vertex- or face-multiplicities of 14 ‘cantankerous’ non-orientable regular maps. Also some discoveries are announced about the genus spectrum of orientably-regular but chiral maps, and the genus spectrum of orientably-regular maps having no multiple edges, made possible by observations of patterns in the extension of these lists to higher genera