Section 15 Factor-group computation and simple groups Instructor: - PowerPoint PPT Presentation

Factor-group computation Simple groups Section 15 – Factor-group computation and simple groups Instructor: Yifan Yang Fall 2006 Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups Outline Factor-group computation 1 Simple groups 2 Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups The problem Problem Given a factor group G / H , find an isomorphic group G ′ that is more familiar to us (so that to study the structure of G / H , we can work on the more familiar group G ′ instead). Some simple facts G / { e } is isomorphic to G itself. 1 G / G is isomorphic to the trivial group with one element. 2 If G is cyclic, then G / H is also cyclic. 3 If G = G 1 × G 2 , then G / ( { e 1 } × G 2 ) is isomorphic to G 1 . 4 If G = G 1 × G 2 , H 1 � G 1 , and H 2 � G 2 , then G / ( H 1 × H 2 ) is 5 isomorphic to ( G 1 / H 1 ) × ( G 2 / H 2 ) . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups The problem Problem Given a factor group G / H , find an isomorphic group G ′ that is more familiar to us (so that to study the structure of G / H , we can work on the more familiar group G ′ instead). Some simple facts G / { e } is isomorphic to G itself. 1 G / G is isomorphic to the trivial group with one element. 2 If G is cyclic, then G / H is also cyclic. 3 If G = G 1 × G 2 , then G / ( { e 1 } × G 2 ) is isomorphic to G 1 . 4 If G = G 1 × G 2 , H 1 � G 1 , and H 2 � G 2 , then G / ( H 1 × H 2 ) is 5 isomorphic to ( G 1 / H 1 ) × ( G 2 / H 2 ) . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups The problem Problem Given a factor group G / H , find an isomorphic group G ′ that is more familiar to us (so that to study the structure of G / H , we can work on the more familiar group G ′ instead). Some simple facts G / { e } is isomorphic to G itself. 1 G / G is isomorphic to the trivial group with one element. 2 If G is cyclic, then G / H is also cyclic. 3 If G = G 1 × G 2 , then G / ( { e 1 } × G 2 ) is isomorphic to G 1 . 4 If G = G 1 × G 2 , H 1 � G 1 , and H 2 � G 2 , then G / ( H 1 × H 2 ) is 5 isomorphic to ( G 1 / H 1 ) × ( G 2 / H 2 ) . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups The problem Problem Given a factor group G / H , find an isomorphic group G ′ that is more familiar to us (so that to study the structure of G / H , we can work on the more familiar group G ′ instead). Some simple facts G / { e } is isomorphic to G itself. 1 G / G is isomorphic to the trivial group with one element. 2 If G is cyclic, then G / H is also cyclic. 3 If G = G 1 × G 2 , then G / ( { e 1 } × G 2 ) is isomorphic to G 1 . 4 If G = G 1 × G 2 , H 1 � G 1 , and H 2 � G 2 , then G / ( H 1 × H 2 ) is 5 isomorphic to ( G 1 / H 1 ) × ( G 2 / H 2 ) . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups The problem Problem Given a factor group G / H , find an isomorphic group G ′ that is more familiar to us (so that to study the structure of G / H , we can work on the more familiar group G ′ instead). Some simple facts G / { e } is isomorphic to G itself. 1 G / G is isomorphic to the trivial group with one element. 2 If G is cyclic, then G / H is also cyclic. 3 If G = G 1 × G 2 , then G / ( { e 1 } × G 2 ) is isomorphic to G 1 . 4 If G = G 1 × G 2 , H 1 � G 1 , and H 2 � G 2 , then G / ( H 1 × H 2 ) is 5 isomorphic to ( G 1 / H 1 ) × ( G 2 / H 2 ) . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups The problem Problem Given a factor group G / H , find an isomorphic group G ′ that is more familiar to us (so that to study the structure of G / H , we can work on the more familiar group G ′ instead). Some simple facts G / { e } is isomorphic to G itself. 1 G / G is isomorphic to the trivial group with one element. 2 If G is cyclic, then G / H is also cyclic. 3 If G = G 1 × G 2 , then G / ( { e 1 } × G 2 ) is isomorphic to G 1 . 4 If G = G 1 × G 2 , H 1 � G 1 , and H 2 � G 2 , then G / ( H 1 × H 2 ) is 5 isomorphic to ( G 1 / H 1 ) × ( G 2 / H 2 ) . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups The problem Problem Given a factor group G / H , find an isomorphic group G ′ that is more familiar to us (so that to study the structure of G / H , we can work on the more familiar group G ′ instead). Some simple facts G / { e } is isomorphic to G itself. 1 G / G is isomorphic to the trivial group with one element. 2 If G is cyclic, then G / H is also cyclic. 3 If G = G 1 × G 2 , then G / ( { e 1 } × G 2 ) is isomorphic to G 1 . 4 If G = G 1 × G 2 , H 1 � G 1 , and H 2 � G 2 , then G / ( H 1 × H 2 ) is 5 isomorphic to ( G 1 / H 1 ) × ( G 2 / H 2 ) . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups Proof of (3)–(5) Proof of (3). If G = � a � , then every coset of H is of the form a k H = ( aH ) k for some integer k . Thus, G / H = � aH � . Proof of (4). Let φ : G 1 × G 2 → G 1 be defined by φ ( g 1 , g 2 ) = g 1 (the projection map). It is easy to see that φ is a homomorphism with Ker ( φ ) = { ( e 1 , g 2 ) : g 2 ∈ G 2 } = { e 1 } × G 2 and Im ( φ ) = G 1 . Then by Theorem 14.11, G / ( { e 1 } × G 2 ) ≃ G 1 . Proof of (5). Let φ : G 1 × G 2 → ( G 1 / H 1 ) × ( G 2 / H 2 ) be defined by φ ( g 1 , g 2 ) = ( g 1 H 1 , g 2 H 2 ) . Then apply Theorem 14.11. Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups Proof of (3)–(5) Proof of (3). If G = � a � , then every coset of H is of the form a k H = ( aH ) k for some integer k . Thus, G / H = � aH � . Proof of (4). Let φ : G 1 × G 2 → G 1 be defined by φ ( g 1 , g 2 ) = g 1 (the projection map). It is easy to see that φ is a homomorphism with Ker ( φ ) = { ( e 1 , g 2 ) : g 2 ∈ G 2 } = { e 1 } × G 2 and Im ( φ ) = G 1 . Then by Theorem 14.11, G / ( { e 1 } × G 2 ) ≃ G 1 . Proof of (5). Let φ : G 1 × G 2 → ( G 1 / H 1 ) × ( G 2 / H 2 ) be defined by φ ( g 1 , g 2 ) = ( g 1 H 1 , g 2 H 2 ) . Then apply Theorem 14.11. Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups Proof of (3)–(5) Proof of (3). If G = � a � , then every coset of H is of the form a k H = ( aH ) k for some integer k . Thus, G / H = � aH � . Proof of (4). Let φ : G 1 × G 2 → G 1 be defined by φ ( g 1 , g 2 ) = g 1 (the projection map). It is easy to see that φ is a homomorphism with Ker ( φ ) = { ( e 1 , g 2 ) : g 2 ∈ G 2 } = { e 1 } × G 2 and Im ( φ ) = G 1 . Then by Theorem 14.11, G / ( { e 1 } × G 2 ) ≃ G 1 . Proof of (5). Let φ : G 1 × G 2 → ( G 1 / H 1 ) × ( G 2 / H 2 ) be defined by φ ( g 1 , g 2 ) = ( g 1 H 1 , g 2 H 2 ) . Then apply Theorem 14.11. Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups Examples Example 1. Let G = R be the additive group of real numbers, and H = n R = { nr : r ∈ R } . What is the group structure of G / H ? Solution. The subgroup H is in fact the whole group G , since every real number x is equal to n ( x / n ) . Thus, R / n R is the trivial group of one element. Example 2. Let G = Z 12 and H = � 4 � = { 0 , 4 , 8 } . What is the group structure of G / H ? Solution. The group Z 12 is cyclic. Thus, G / H is cyclic of order ( G : H ) = 4, i.e., G / H ≃ Z 4 . In fact, G / H = { H , 1 + H , 2 + H , 3 + H } = � 1 + H � . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups Examples Example 1. Let G = R be the additive group of real numbers, and H = n R = { nr : r ∈ R } . What is the group structure of G / H ? Solution. The subgroup H is in fact the whole group G , since every real number x is equal to n ( x / n ) . Thus, R / n R is the trivial group of one element. Example 2. Let G = Z 12 and H = � 4 � = { 0 , 4 , 8 } . What is the group structure of G / H ? Solution. The group Z 12 is cyclic. Thus, G / H is cyclic of order ( G : H ) = 4, i.e., G / H ≃ Z 4 . In fact, G / H = { H , 1 + H , 2 + H , 3 + H } = � 1 + H � . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups

Factor-group computation Simple groups Examples Example 1. Let G = R be the additive group of real numbers, and H = n R = { nr : r ∈ R } . What is the group structure of G / H ? Solution. The subgroup H is in fact the whole group G , since every real number x is equal to n ( x / n ) . Thus, R / n R is the trivial group of one element. Example 2. Let G = Z 12 and H = � 4 � = { 0 , 4 , 8 } . What is the group structure of G / H ? Solution. The group Z 12 is cyclic. Thus, G / H is cyclic of order ( G : H ) = 4, i.e., G / H ≃ Z 4 . In fact, G / H = { H , 1 + H , 2 + H , 3 + H } = � 1 + H � . Instructor: Yifan Yang Section 15 – Factor-group computation and simple groups