Section 13 – Homomorphisms Instructor: Yifan Yang Fall 2006 Instructor: Yifan Yang Section 13 – Homomorphisms
Homomorphisms Definition A map φ of a group G into a group G ′ is a homomorphism if φ ( ab ) = φ ( a ) φ ( b ) for all a , b ∈ G . Instructor: Yifan Yang Section 13 – Homomorphisms
Examples Let φ : G → G ′ be defined by φ ( g ) = e ′ for all g ∈ G . Then 1 clearly, φ ( ab ) = e ′ = e ′ e ′ = φ ( a ) φ ( b ) for all a , b ∈ G . This is called the trivial homomorphism. Let φ : Z → Z be defined by φ ( n ) = 2 n for all n ∈ Z . Then φ 2 is a homomorphism. Let S n be the symmetric group on n letters, and let 3 φ : S n → Z 2 be defined by � 0 , if σ is an even permutation , φ ( σ ) = 1 , if σ is an odd permutation . Then φ is a homomorphism. (Check case by case.) Instructor: Yifan Yang Section 13 – Homomorphisms
Examples Let φ : G → G ′ be defined by φ ( g ) = e ′ for all g ∈ G . Then 1 clearly, φ ( ab ) = e ′ = e ′ e ′ = φ ( a ) φ ( b ) for all a , b ∈ G . This is called the trivial homomorphism. Let φ : Z → Z be defined by φ ( n ) = 2 n for all n ∈ Z . Then φ 2 is a homomorphism. Let S n be the symmetric group on n letters, and let 3 φ : S n → Z 2 be defined by � 0 , if σ is an even permutation , φ ( σ ) = 1 , if σ is an odd permutation . Then φ is a homomorphism. (Check case by case.) Instructor: Yifan Yang Section 13 – Homomorphisms
Examples Let φ : G → G ′ be defined by φ ( g ) = e ′ for all g ∈ G . Then 1 clearly, φ ( ab ) = e ′ = e ′ e ′ = φ ( a ) φ ( b ) for all a , b ∈ G . This is called the trivial homomorphism. Let φ : Z → Z be defined by φ ( n ) = 2 n for all n ∈ Z . Then φ 2 is a homomorphism. Let S n be the symmetric group on n letters, and let 3 φ : S n → Z 2 be defined by � 0 , if σ is an even permutation , φ ( σ ) = 1 , if σ is an odd permutation . Then φ is a homomorphism. (Check case by case.) Instructor: Yifan Yang Section 13 – Homomorphisms
example Let GL ( n , R ) be the set of all invertible n × n matrices over 1 R . Define φ : GL ( n , R ) → R × by φ ( A ) = det ( A ) . Then φ is a homomorphism since det ( AB ) = det ( A ) det ( B ) . Let F be the additive group of all polynomials with real 2 coefficients. For a given real number a , the function φ a : F → R defined by φ ( f ) = f ( a ) is a homomorphism, called an evaluation homomorphism. Let n be a positive integer. Define φ n : Z → Z n by 3 φ n ( r ) = ¯ r . Then φ n is a homomorphism. Let G = G 1 × G 2 × . . . × G n be a direct product of groups. 4 The projection map π i : G → G i defined by π i ( a 1 , a 2 , . . . , a i , . . . , a n ) = a i is a homomorphism. Instructor: Yifan Yang Section 13 – Homomorphisms
example Let GL ( n , R ) be the set of all invertible n × n matrices over 1 R . Define φ : GL ( n , R ) → R × by φ ( A ) = det ( A ) . Then φ is a homomorphism since det ( AB ) = det ( A ) det ( B ) . Let F be the additive group of all polynomials with real 2 coefficients. For a given real number a , the function φ a : F → R defined by φ ( f ) = f ( a ) is a homomorphism, called an evaluation homomorphism. Let n be a positive integer. Define φ n : Z → Z n by 3 φ n ( r ) = ¯ r . Then φ n is a homomorphism. Let G = G 1 × G 2 × . . . × G n be a direct product of groups. 4 The projection map π i : G → G i defined by π i ( a 1 , a 2 , . . . , a i , . . . , a n ) = a i is a homomorphism. Instructor: Yifan Yang Section 13 – Homomorphisms
example Let GL ( n , R ) be the set of all invertible n × n matrices over 1 R . Define φ : GL ( n , R ) → R × by φ ( A ) = det ( A ) . Then φ is a homomorphism since det ( AB ) = det ( A ) det ( B ) . Let F be the additive group of all polynomials with real 2 coefficients. For a given real number a , the function φ a : F → R defined by φ ( f ) = f ( a ) is a homomorphism, called an evaluation homomorphism. Let n be a positive integer. Define φ n : Z → Z n by 3 φ n ( r ) = ¯ r . Then φ n is a homomorphism. Let G = G 1 × G 2 × . . . × G n be a direct product of groups. 4 The projection map π i : G → G i defined by π i ( a 1 , a 2 , . . . , a i , . . . , a n ) = a i is a homomorphism. Instructor: Yifan Yang Section 13 – Homomorphisms
example Let GL ( n , R ) be the set of all invertible n × n matrices over 1 R . Define φ : GL ( n , R ) → R × by φ ( A ) = det ( A ) . Then φ is a homomorphism since det ( AB ) = det ( A ) det ( B ) . Let F be the additive group of all polynomials with real 2 coefficients. For a given real number a , the function φ a : F → R defined by φ ( f ) = f ( a ) is a homomorphism, called an evaluation homomorphism. Let n be a positive integer. Define φ n : Z → Z n by 3 φ n ( r ) = ¯ r . Then φ n is a homomorphism. Let G = G 1 × G 2 × . . . × G n be a direct product of groups. 4 The projection map π i : G → G i defined by π i ( a 1 , a 2 , . . . , a i , . . . , a n ) = a i is a homomorphism. Instructor: Yifan Yang Section 13 – Homomorphisms
Properties of homomorphisms Definition Let φ be a mapping of a set X into a set Y . Let A ⊂ X and B ⊂ Y . The image φ [ A ] of A under φ is { φ ( a ) : a ∈ A } . The set φ [ X ] is the range of φ . The inverse image φ − 1 [ B ] of B in X is { x ∈ X : φ ( x ) ∈ B } . Instructor: Yifan Yang Section 13 – Homomorphisms
Properties of homomorphisms Theorem (13.12) Let φ be a homomorphism of a group G into a group G ′ . φ ( e ) = e ′ . 1 φ ( a − 1 ) = φ ( a ) − 1 for all a ∈ G. 2 If H is a subgroup of G, then φ [ H ] is a subgroup of G ′ . 3 If K ′ is a subgroup of G ′ , then φ − 1 [ K ′ ] is a subgroup of G. 4 Instructor: Yifan Yang Section 13 – Homomorphisms
Properties of homomorphisms Theorem (13.12) Let φ be a homomorphism of a group G into a group G ′ . φ ( e ) = e ′ . 1 φ ( a − 1 ) = φ ( a ) − 1 for all a ∈ G. 2 If H is a subgroup of G, then φ [ H ] is a subgroup of G ′ . 3 If K ′ is a subgroup of G ′ , then φ − 1 [ K ′ ] is a subgroup of G. 4 Instructor: Yifan Yang Section 13 – Homomorphisms
Properties of homomorphisms Theorem (13.12) Let φ be a homomorphism of a group G into a group G ′ . φ ( e ) = e ′ . 1 φ ( a − 1 ) = φ ( a ) − 1 for all a ∈ G. 2 If H is a subgroup of G, then φ [ H ] is a subgroup of G ′ . 3 If K ′ is a subgroup of G ′ , then φ − 1 [ K ′ ] is a subgroup of G. 4 Instructor: Yifan Yang Section 13 – Homomorphisms
Properties of homomorphisms Theorem (13.12) Let φ be a homomorphism of a group G into a group G ′ . φ ( e ) = e ′ . 1 φ ( a − 1 ) = φ ( a ) − 1 for all a ∈ G. 2 If H is a subgroup of G, then φ [ H ] is a subgroup of G ′ . 3 If K ′ is a subgroup of G ′ , then φ − 1 [ K ′ ] is a subgroup of G. 4 Instructor: Yifan Yang Section 13 – Homomorphisms
Properties of homomorphisms Theorem (13.12) Let φ be a homomorphism of a group G into a group G ′ . φ ( e ) = e ′ . 1 φ ( a − 1 ) = φ ( a ) − 1 for all a ∈ G. 2 If H is a subgroup of G, then φ [ H ] is a subgroup of G ′ . 3 If K ′ is a subgroup of G ′ , then φ − 1 [ K ′ ] is a subgroup of G. 4 Instructor: Yifan Yang Section 13 – Homomorphisms
Proof of Theorem 13.12 Proof of φ ( e ) = e ′ . Consider φ ( a ) , where a ∈ G . We have φ ( a ) = φ ( ae ) = φ ( a ) φ ( e ) . By the cancellation law, φ ( e ) must equal to the identity e ′ . Proof of φ ( a − 1 ) = φ ( a ) − 1 . We have φ ( a ) φ ( a − 1 ) = φ ( aa − 1 ) = φ ( e ) = e ′ . Thus, φ ( a − 1 ) = φ ( a ) − 1 . Instructor: Yifan Yang Section 13 – Homomorphisms
Proof of Theorem 13.12 Proof of φ ( e ) = e ′ . Consider φ ( a ) , where a ∈ G . We have φ ( a ) = φ ( ae ) = φ ( a ) φ ( e ) . By the cancellation law, φ ( e ) must equal to the identity e ′ . Proof of φ ( a − 1 ) = φ ( a ) − 1 . We have φ ( a ) φ ( a − 1 ) = φ ( aa − 1 ) = φ ( e ) = e ′ . Thus, φ ( a − 1 ) = φ ( a ) − 1 . Instructor: Yifan Yang Section 13 – Homomorphisms
Proof of Theorem 13.12 Proof of Theorem 13.12(3). We need to prove Closed: Suppose that a ′ , b ′ ∈ φ [ H ] . Then there exist 1 a , b ∈ H such that φ ( a ) = a ′ and φ ( b ) = b ′ . Thus, a ′ b ′ = φ ( a ) φ ( b ) = φ ( ab ) . Since H is a subgroup, ab ∈ H . Therefore, a ′ b ′ is in φ [ H ] . identity: By Part (1), e ′ = φ ( e ) ∈ φ [ H ] . 2 inverse: Suppose that a ′ ∈ φ [ H ] . Then a ′ = φ ( a ) for some 3 a ∈ H . By Part (b), ( a ′ ) − 1 = φ ( a ) − 1 = φ ( a − 1 ) , and thus ( a ′ ) − 1 ∈ φ [ H ] . Instructor: Yifan Yang Section 13 – Homomorphisms
Proof of Theorem 13.12 Proof of Theorem 13.12(3). We need to prove Closed: Suppose that a ′ , b ′ ∈ φ [ H ] . Then there exist 1 a , b ∈ H such that φ ( a ) = a ′ and φ ( b ) = b ′ . Thus, a ′ b ′ = φ ( a ) φ ( b ) = φ ( ab ) . Since H is a subgroup, ab ∈ H . Therefore, a ′ b ′ is in φ [ H ] . identity: By Part (1), e ′ = φ ( e ) ∈ φ [ H ] . 2 inverse: Suppose that a ′ ∈ φ [ H ] . Then a ′ = φ ( a ) for some 3 a ∈ H . By Part (b), ( a ′ ) − 1 = φ ( a ) − 1 = φ ( a − 1 ) , and thus ( a ′ ) − 1 ∈ φ [ H ] . Instructor: Yifan Yang Section 13 – Homomorphisms
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