College Algebra Exam Review 106

College Algebra Exam Review 106 - ³ is said to be odd The...

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116 2. BASIC THEORY OF GROUPS Example 2.4.18. Let G be any group and a 2 G . If the order of a is n, then the kernel of the homomorphism k 7! a k from Z to G is the set of all multiples of n , f kn W k 2 Z g . If a is of infinite order, then the kernel of the homomorphism is f 0 g . Example 2.4.19. In particular, the kernel of the homomorphism from Z to Z n defined by k 7! ŒkŁ is Œ0Ł D f kn W k 2 Z g . Example 2.4.20. If G is an abelian group and n is a fixed integer, then the kernel of the homomorphism g 7! g n from G to G is the set of elements whose order divides n . Parity of Permutations Additional examples of homomorphisms are explored in the Exercises. In particular, it is shown in the Exercises that there is a homomorphisms ± W S n ! f˙ 1 g with the property that ±.²/ D ± 1 for any 2–cycle ² . This is an example of a homomorphism that is very far from being injective and that picks out an essential structural feature of the symmetric group. Definition 2.4.21. The homomorphism ± is called the sign (or parity ) ho- momorphism. A permutation ³ is said to be even if ±.³/ D 1 , that is, if ³ is in the kernel of the sign homomorphism. Otherwise,
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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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