Abstract

Abstract. A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decomposition of the group algebra into irreducible factors. This gives a new proof of the classical formulas of Frobenius, Schur, and Mednykh. Let S be a compact oriented or non-orientable surface without boundary, and χ(S) its Euler characteristic. The subject of our study is a generating function of the number |Hom(π1(S), G) | of homomorphisms from the fundamental group of S into a finite group G. We give a generating function in terms of a non-commutative integral Eqn.(2.7) or Eqn.(3.2),

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