Section VI in a Nutshell

Section VI in a Nutshell We examine more deeply the permutations of the symmetric groups Sn. A transposition

is a cycle of length two. Any permutation can be factored as a product of transpositions (Theorem 27.1), and while this factorization is not unique, all factorizations of a given permutation have the same parity; thus we can classify a permutation as either even or odd. The set An of even permutations is a group called the nth alternating group. The alternating group An is a normal subgroup of the symmetric group Sn and [Sn : An] = 2. Furthermore, An is simple for n 6= 4.