Abstract
Given a permutation s on a finite set Ω of order n, define c(s) to be the number of cycles of sand Ind(s) = n - c(s). Define a genus g system to be a triple ( G, Ω, S), where Ω is a finite set, G is a transitive subgroup of Sym(Ω), and S = (g_j: 1 ⩽j⩽r is a family of elements of G^# such that G = ⟨S⟩, g_1...g_r = 1, and 2(❘Ω❘ + g-1)= ∑_(j=1) Ind(g_j).
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