A theorem on permutations

AbstractOut of the classical theory of Riemann surfaces, we extract the following combinatorial theorem: If π1,…, πm are permutations of 1, 2,…, n, such that π1…πm = 1, then v(π1) + … + v(πm) ⩾ 2(n − t), where v(π) = n − r, r being the number of orbits of the cyclic group generated by the permutation π acting on the set {1,…,n}, and where t is the number of orbits of the group generated by π1,…,πm.The direct proof of this theorem seems to be difficult