{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 106

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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,
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online