1,299 research outputs found
Primitive permutation groups containing a cycle
The primitive finite permutation groups containing a cycle are classified. Of
these, only the alternating and symmetric groups contain a cycle fixing at
least three points. The contributions of Jordan and Marggraff to this topic are
briefly discussed.Comment: 6 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
Unstable maps
A map which is non-orientable or has non-empty boundary has a canonical
double cover which is orientable and has empty boundary. The map is called
stable if every automorphism of this cover is a lift of an automorphism of the
map. This note describes several infinite families of unstable maps, and
relates them to similar phenomena for graphs, hypermaps and Klein surfaces.Comment: 11 pages, 4 figure
Classification and Galois conjugacy of Hamming maps
We show that for each d>0 the d-dimensional Hamming graph H(d,q) has an
orientably regular surface embedding if and only if q is a prime power p^e. If
q>2 there are up to isomorphism \phi(q-1)/e such maps, all constructed as
Cayley maps for a d-dimensional vector space over the field of order q. We show
that for each such pair d, q the corresponding Belyi pairs are conjugate under
the action of the absolute Galois group, and we determine their minimal field
of definition. We also classify the orientably regular embedding of merged
Hamming graphs for q>3
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