In mathematics, Hurwitz's automorphisms theorem bounds the group of automorphisms, via conformal mappings, of a compact Riemann surface of genus g > 1, telling us that the order of the group of such automorphisms is bounded by History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time  days, seasons, years. ...
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
In mathematics, a mapping w = f(z) is anglepreserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ...
In mathematics, particularly in complex analysis, a Riemann surface is a onedimensional complex manifold. ...
In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
In mathematics, a finite group is a group which has finitely many elements. ...
 84(g − 1).
A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because of the equivalence of categories between compact Riemann surfaces and complex projective curves, a Hurwitz surface can also be called a Hurwitz curve. The theorem is due to Adolf Hurwitz, who proved it in 1893. In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ...
In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...
Adolf Hurwitz Adolf Hurwitz (26 March 1859 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to JeanPierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim...
The conformal mappings of the Hurwitz surface correspond to orientationpreserving isogenies of the hyperbolic plane. In order to make the automorphism group as large as possible, we want the area of a triangular fundamental region to be as small as possible, which means we want A triangle immersed in a saddleshape plane, as well as two diverging parallel lines. ...
 π(1 − 1/p − 1/q − 1/r)
to be as small as possible, where p, q, and r are positive integers defining the vertex angles π/p, π/q and π/r of a fundamental region for a tiling of the hyperbolic plane. Asking for integers which make In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a wellbehaved set topologically, in a sense that can be made precise in one of several ways. ...
In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and threedimensional objects are considered as well. ...
 1 − 1/p − 1/q − 1/r
positive and as small as possible is a Diophantine question; to which the answer is  1 − 1/2 − 1/3 − 1/7 = 1/42.
Since a reflection flips the triangle, we join two of them and obtaining the orientationpreserving tiling polygon. A Hurwitz group is characterized by the property that it is a finite group with generators a and b and relations including  a^{2} = b^{3} = (ab)^{7} = 1;
in other words it is a finite group generated by two elements of orders two and three, whose product is of order seven. Hurwitz's result was that we obtain a Hurwitz surface, with the automorphisms maximum achieved, if and only if the automorphism group is a Hurwitz group. The smallest Hurwitz group is the special linear group L_{2}(7), of order 168, and the corresponding curve is the Klein quartic curve. The projective special linear group G = PSL(2,7) is a finite group in mathematics that has important applications in algebra, geometry, and number theory. ...
The Klein quartic x3y + y3z + z3x = 0, named after Felix Klein, is a Riemann surface, and a curve of genus 3 over the complex numbers C. The Klein quartic has automorphism group isomorphic to the projective special linear group G = PSL(2,7). ...
Next is the Macbeath curve, with automorphism group L_{2}(8) of order 504. Many more finite simple groups are Hurwitz groups; for instance all but 64 of the alternating groups are Hurwitz groups, the largest nonHurwitz example being of degree 167. In mathematics an alternating group is the group of even permutations of a finite set. ...
The sporadic Hurwitz groups are the Janko groups J_{1}, J_{2} and J_{4}, the Fischer groups Fi_{22} and Fi'_{24}, the Rudvalis group, the Held group, the Thompson group, the HaradaNorton group,the third Conway group Co_{3}, the Lyons group and best of all, the Monster. The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...
In mathematics, the Janko groups J1, J2, J3 and J4 are four of the twentysix sporadic groups; their respective orders are: J1 The smallest Janko group, J1 of order 175560, has a presentation in terms of two generators a and b and c = abab1 as It can also...
In mathematics, the term Fischer groups usually refers to the three finite groups denoted Fi22, Fi23, and Fi24, all of which are simple groups, and constitute three of the 26 sporadic groups. ...
The Rudvalis group, Ru, is the sporadic simple group of order . It is named for Arunas Rudvalis. ...
In mathematics, the Held group, He, is the unique finite simple sporadic group of order . ...
In mathematics, the term Thompson group or Thompsons group can refer to either the sporadic finite simple group Th  see Thompson group (finite); one of the three infinite groups F, T and V first studied by the logician Richard Thompson in 1965  see Thompson groups. ...
In mathematics, the HaradaNorton group, HN is the sporadic simple group of order . It is named for Koichiro Harada and Simon Norton. ...
In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway. ...
In mathematics, the Lyons group, Ly, is a finite sporadic simple group of order It can be characterized as the unique simple group where the centralizer of an involution, and hence of all the involutions, is isomorphic to the nontrivial central extension of the cyclic group C2 by the alternating...
In mathematics, the Monster group M is a group of order 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ...
