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algebra-I-notes

# algebra-I-notes - Abstract Algebra I Notes UIUC MATH 500...

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Abstract Algebra I Notes UIUC MATH 500, F’08 Jingjin Yu 1 Groups 08/25/08 - 08/27/08 Definition 1.1 Semigroups, monoids, groups, rings and commutative rings. For a map G G G , consider the following properties: 1) Closure, 2) Association, 3) Identity, 4) Inverse, 5) Commutative. A set G with a composition law G G G is called : a semigroup if it satisfies 1-2, a monoid if it satisfies 1-3, a group if it satisfies 1-4, a abelian group if it satisfies 1-5. Definition 1.2 A subgroup H of group G is a subset of G that is also a group with the same map of G G G . Example 1.3 0: semigroup; 0: monoid; : group. Definition 1.4 A group homomorphism is a function f : G H s.t. f xy f x f y . If f is injective, surjective, and bijective, then we call them monomorphism , epimorphism , isomorphism . Example 1.5 , , is an injective homomorphism; but there does not exist a nonzero homomor- phism , , . Suppose not then f 1 0 and nf 1 n f n 1 n f 1 f 1 n 0. But no matter what f 1 is, it cannot be infinity and there are some n such that 0 f 1 n f 1 n 1. This is not possible since f 1 n . Definition 1.6 A permutation of a set X is a bijection X X : 1 . . . . . . n a 1 . . . . . . a n 1

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Definition 1.7 The symmetric group on a set X is the collection of all permutations on X with notation Symm X . If X 1 , . . . , n , we write S n for Symm X . Remark. We have injection φ : S n S n 1 by extending any permutation f with f n 1 n 1. We may define S n 1 S n . If we let X 1 , 2 , . . . , then S Symm X since the permutation σ 2 , 1 , 4 , 3 , 6 , 5 . . . Symm X , but σ S . Theorem 1.8 Let f : G H be a homomorphism. (1) f is a monomorphism if and only if ker f e . (2) f is an isomorphism if and only if there eixsts a homomorphism f 1 s.t. f f 1 1 H , f 1 f 1 G . Proof. (1) ( ) Clear. ( ) Suppose f x f y , then f xy 1 f 1 1 H , by assumption xy 1 1 x y . (2) ( ) f is invertible, we just need to verify that it is a homomorphism: f 1 ab f 1 a f 1 b . ( ) Clear. 08/29/08 Definition 1.9 let a 1 , . . . , a r be distinct elements of X 1 , 2 , . . . , n , and let Y X a 1 , . . . , a r . If f S n fixes every element in Y and f a i a i 1 for i 1 , r 1 and f a r a 1 , then f is called an r -cycle and we write f a 1 , . . . , a r . A 2-cycle is called a transposition . Definition 1.10 Two permutations α, β are disjoint if (1) α k k β k k , and (2) β k k α k k . Proposition 1.11 Every permutation α S n is a composite (product) of disjoint cycles. Proof. By induction on the number of elements that moved by α . (case m 0) α 1 is the identity. (case m 0) Choose a 1 with α a 1 a 1 and let a 2 α a 1 , . . . a i 1 α a i until we have a l a 1 , . . . , a l 1 . We claim that a l a 1 . Suppose not and a l a i , 1 i l , then a l α a l 1 and a i α a i 1 , which imply that a l 1 a i 1 , contradicting that a l is the first repetition. We may then get a cycle that is disjoint from the rest of the permutation and apply induction hypothesis. Definition 1.12 G is a group and a, g G , then gag 1 is a conjugate of a . Example 1.13 G S n , let α a 1 , . . . , a n S n , then for τ S n , τατ 1 τ a 1 , . . . , τ a n .
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algebra-I-notes - Abstract Algebra I Notes UIUC MATH 500...

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