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Register a new userKleins ten planar dessins of degree 11, and beyond
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We reinterpret ideas in Kleins paper on transformations of degree~$11$ from the modern point of view of dessins denfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman--Horn Conjecture and extensive computer searches to support a conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for some prime power $q$, in which case infinitely many groups ${rm PSL}_n(q)$ arise as permutation groups and monodromy groups of degree $p$ (an open problem in group theory).
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This is an English translation of Felix Kleins paper Ueber die Transformation elfter Ordnung der elliptischen Functionen from 1879.
A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism group of an object is the centraliser of its monodromy group. An alternative form of the theorem, valid for finite objects, is discussed, with counterexamples based on Baumslag--Solitar groups to show how it fails more generally. The automorphism groups of objects with primitive monodromy groups are described, as are those of non-connected objects.
It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin denfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compact dessins. Moreover, we show that any tame action of a countable group is so realizable.
The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins denfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data.
The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the $195,826,352$ conjugacy classes of transitive subgroups of the symmetric group $S_{48}$ of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive graphs of degree $48$, of which there are $1,538,868,366$, all but $0.1625%$ of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published by the third author as a separate paper
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