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Now a days, the theory of Riemann surfaces occupies a very special place in mathematics. The basic idea of a Riemann surface is that it is a space which, locally, looks just like an open set in complex plane. First we start preliminaries from set topology and complex analysis, and then we will deal with the construction part of Riemann surface. Next we will present the survey of function theory in the complex plane, the definition of holomorphic and meromorphic functions in Riemann surface. After that we will give a bound to the cardinality of the group, depending on the genus, which act holomorphically and effectively on a compact Riemann surface of genus greater then equal to two. This bound will be obtained from Hurwitz's theorem which can be generalized to be an upper bound to the cardinality of the automorphism group of any Riemann surface of genus greater than equal to two.