Unformatted text preview: ³ is said to be odd . The subgroup of even permutations (that is, the kernel of ± ) is generally denoted A n . This subgroup is also referred to as the alternating group . The following statement about even and odd permutations is implicit in the Exercises: Proposition 2.4.22. A permutation ³ is even if, and only if, ³ can be written as a product of an even number of 2–cycles. Even and odd permutations have the following property: The product of two even permutations is even; the product of an even and an odd per-mutation is odd, and the product of two odd permutations is even. Hence Corollary 2.4.23. The set of odd permutations in S n is .12/A n , where A n denotes the subgroup of even permutations. Proof. .12/A n is contained in the set of odd permutations. But if ´ is any odd permutation, then .12/´ is even, so ´ D .12/. .12/´/ 2 .12/A n . n...
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- Fall '08