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For (n\ge 5), consider the symmetric group (S_n) of all permutations of ({1,2,\dots ,n}) and its alternating subgroup (A_n) consisting of all even permutations. The index of (A_n) in (S_n) is 2, so the quotient group (S_n/A_n) has exactly two elements, namely the two cosets [ A_n \quad\text{and}\quad \sigma A_n, ] where (\sigma) is any fixed odd permutation, for example the transposition __________ ________ _______ ________ _______ ______ ________ __________ _________.
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