768 research outputs found
Enumerating Regular Objects associated with Suzuki Groups
We use the M\"obius function of the simple Suzuki group Sz(q) to enumerate
regular objects such as maps, hypermaps, dessins d'enfants and surface
coverings with automorphism groups isomorphic to Sz(q).Comment: 20 page
Regular dessins with a given automorphism group
Dessins d'enfants are combinatorial structures on compact Riemann surfaces
defined over algebraic number fields, and regular dessins are the most
symmetric of them. If G is a finite group, there are only finitely many regular
dessins with automorphism group G. It is shown how to enumerate them, how to
represent them all as quotients of a single regular dessin U(G), and how
certain hypermap operations act on them. For example, if G is a cyclic group of
order n then U(G) is a map on the Fermat curve of degree n and genus
(n-1)(n-2)/2. On the other hand, if G=A_5 then U(G) has genus
274218830047232000000000000000001. For other non-abelian finite simple groups,
the genus is much larger.Comment: 19 page
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