2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64821A 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.12 pages, 1 figure. Prepared in AMS-LaTeXQuantum AlgebraMathematical Physicstext