Editing Hurwitz's automorphisms theorem (section)

{{short description|Bounds the order of the group of automorphisms of a compact Riemann surface of genus g > 1}}
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In [[mathematics]], '''Hurwitz's automorphisms theorem''' bounds the order of the group of [[automorphism]]s, via [[orientation-preserving]] [[conformal mapping]]s, of a compact [[Riemann surface]] of [[genus (mathematics)|genus]] ''g'' > 1, stating that the number of such automorphisms cannot exceed 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 compact Riemann surfaces are synonymous with non-singular [[algebraic curve|complex projective algebraic curves]], a Hurwitz surface can also be called a '''Hurwitz curve'''.<ref>Technically speaking, there is an [[equivalence of categories]] between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms.</ref> The theorem is named after [[Adolf Hurwitz]], who proved it in {{Harv|Hurwitz|1893}}.

Hurwitz's bound also holds for algebraic curves over a field of characteristic 0, and over fields of positive characteristic ''p'' > 0 for groups whose order is coprime to ''p'', but can fail over fields of positive characteristic ''p'' > 0 when ''p'' divides the group order. For example,  the double cover of the projective line ''y''<sup>2</sup> = ''x<sup>p</sup>'' − ''x'' branched at all points defined over the prime field has genus ''g'' = (''p'' − 1)/2 but is acted on by the group PGL<sub>2</sub>(''p'') of  order ''p''<sup>3</sup> − ''p''.