Direct products Finitely generated abelian groups Section 11 – Direct products and finitely generated abelian groups Instructor: Yifan Yang Fall 2006 Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Outline Direct products 1 Finitely generated abelian groups 2 Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Cartesian product Definition The Cartesian product of sets S 1 , S 2 , . . . , S n is the set of ordered n -tuples ( a 1 , a 2 , . . . , a n ) , where a i ∈ S i for i = 1 , 2 , . . . , n . It is denoted by either S 1 × S 2 × . . . × S n or n � S i . i = 1 Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Direct products Theorem Let G 1 , G 2 , . . . , G n be groups. For ( a 1 , a 2 , . . . , a n ) and ( b 1 , b 2 , . . . , b n ) in � n i = 1 G i define ( a 1 , a 2 , . . . , a n )( b 1 , b 2 , . . . , b n ) to be ( a 1 b 1 , a 2 b 2 , . . . , a n b n ) . Then � n i = 1 G i is a group under this binary operation. Proof. Straightforward. See the textbook. Definition The group � n i = 1 G i defined above is called the direct product of the groups G i . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Direct products Theorem Let G 1 , G 2 , . . . , G n be groups. For ( a 1 , a 2 , . . . , a n ) and ( b 1 , b 2 , . . . , b n ) in � n i = 1 G i define ( a 1 , a 2 , . . . , a n )( b 1 , b 2 , . . . , b n ) to be ( a 1 b 1 , a 2 b 2 , . . . , a n b n ) . Then � n i = 1 G i is a group under this binary operation. Proof. Straightforward. See the textbook. Definition The group � n i = 1 G i defined above is called the direct product of the groups G i . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Direct products Theorem Let G 1 , G 2 , . . . , G n be groups. For ( a 1 , a 2 , . . . , a n ) and ( b 1 , b 2 , . . . , b n ) in � n i = 1 G i define ( a 1 , a 2 , . . . , a n )( b 1 , b 2 , . . . , b n ) to be ( a 1 b 1 , a 2 b 2 , . . . , a n b n ) . Then � n i = 1 G i is a group under this binary operation. Proof. Straightforward. See the textbook. Definition The group � n i = 1 G i defined above is called the direct product of the groups G i . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Direct sums When the groups G i are all abelian with operation + , we sometimes use the notation G 1 ⊕ G 2 · · · ⊕ G n = ⊕ n i = 1 G i instead of G 1 × G 2 × · · · G n = � n i = 1 G i . Also, we call the group ⊕ n i = 1 G i the direct sum of the groups G i . Remark The changing of the order of the factors in a direct product yields a group isomorphic to the original one. For example, define φ : G 1 × G 2 → G 2 × G 1 by φ (( g 1 , g 2 )) = ( g 2 , g 1 ) . It is easy to verify that φ is an isomorphism. Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Direct sums When the groups G i are all abelian with operation + , we sometimes use the notation G 1 ⊕ G 2 · · · ⊕ G n = ⊕ n i = 1 G i instead of G 1 × G 2 × · · · G n = � n i = 1 G i . Also, we call the group ⊕ n i = 1 G i the direct sum of the groups G i . Remark The changing of the order of the factors in a direct product yields a group isomorphic to the original one. For example, define φ : G 1 × G 2 → G 2 × G 1 by φ (( g 1 , g 2 )) = ( g 2 , g 1 ) . It is easy to verify that φ is an isomorphism. Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Direct sums When the groups G i are all abelian with operation + , we sometimes use the notation G 1 ⊕ G 2 · · · ⊕ G n = ⊕ n i = 1 G i instead of G 1 × G 2 × · · · G n = � n i = 1 G i . Also, we call the group ⊕ n i = 1 G i the direct sum of the groups G i . Remark The changing of the order of the factors in a direct product yields a group isomorphic to the original one. For example, define φ : G 1 × G 2 → G 2 × G 1 by φ (( g 1 , g 2 )) = ( g 2 , g 1 ) . It is easy to verify that φ is an isomorphism. Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Example Consider Z 2 × Z 3 . It has 6 elements, namely ( 0 , 0 ) , ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 0 ) , ( 1 , 1 ) , and ( 1 , 2 ) . Since it is abelian, it must be isomorphic to Z 6 . (A group of order 6 is isomorphic to either Z 6 or S 3 .) In fact, if we set g = ( 1 , 1 ) , then we have 2 g = ( 1 , 1 ) + ( 1 , 1 ) = ( 0 , 2 ) 3 g = 2 g + ( 1 , 1 ) = ( 0 , 2 ) + ( 1 , 1 ) = ( 1 , 0 ) 4 g = 3 g + ( 1 , 1 ) = ( 1 , 0 ) + ( 1 , 1 ) = ( 0 , 1 ) 5 g = 4 g + ( 1 , 1 ) = ( 0 , 1 ) + ( 1 , 1 ) = ( 1 , 2 ) 6 g = 5 g + ( 1 , 1 ) = ( 1 , 2 ) + ( 1 , 1 ) = ( 0 , 0 ) . In other words, Z 2 × Z 3 is cyclic of order 6, and hence isomorphic to Z 6 . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Example Consider Z 2 × Z 3 . It has 6 elements, namely ( 0 , 0 ) , ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 0 ) , ( 1 , 1 ) , and ( 1 , 2 ) . Since it is abelian, it must be isomorphic to Z 6 . (A group of order 6 is isomorphic to either Z 6 or S 3 .) In fact, if we set g = ( 1 , 1 ) , then we have 2 g = ( 1 , 1 ) + ( 1 , 1 ) = ( 0 , 2 ) 3 g = 2 g + ( 1 , 1 ) = ( 0 , 2 ) + ( 1 , 1 ) = ( 1 , 0 ) 4 g = 3 g + ( 1 , 1 ) = ( 1 , 0 ) + ( 1 , 1 ) = ( 0 , 1 ) 5 g = 4 g + ( 1 , 1 ) = ( 0 , 1 ) + ( 1 , 1 ) = ( 1 , 2 ) 6 g = 5 g + ( 1 , 1 ) = ( 1 , 2 ) + ( 1 , 1 ) = ( 0 , 0 ) . In other words, Z 2 × Z 3 is cyclic of order 6, and hence isomorphic to Z 6 . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Example Consider Z 2 × Z 3 . It has 6 elements, namely ( 0 , 0 ) , ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 0 ) , ( 1 , 1 ) , and ( 1 , 2 ) . Since it is abelian, it must be isomorphic to Z 6 . (A group of order 6 is isomorphic to either Z 6 or S 3 .) In fact, if we set g = ( 1 , 1 ) , then we have 2 g = ( 1 , 1 ) + ( 1 , 1 ) = ( 0 , 2 ) 3 g = 2 g + ( 1 , 1 ) = ( 0 , 2 ) + ( 1 , 1 ) = ( 1 , 0 ) 4 g = 3 g + ( 1 , 1 ) = ( 1 , 0 ) + ( 1 , 1 ) = ( 0 , 1 ) 5 g = 4 g + ( 1 , 1 ) = ( 0 , 1 ) + ( 1 , 1 ) = ( 1 , 2 ) 6 g = 5 g + ( 1 , 1 ) = ( 1 , 2 ) + ( 1 , 1 ) = ( 0 , 0 ) . In other words, Z 2 × Z 3 is cyclic of order 6, and hence isomorphic to Z 6 . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Example Consider Z 2 × Z 3 . It has 6 elements, namely ( 0 , 0 ) , ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 0 ) , ( 1 , 1 ) , and ( 1 , 2 ) . Since it is abelian, it must be isomorphic to Z 6 . (A group of order 6 is isomorphic to either Z 6 or S 3 .) In fact, if we set g = ( 1 , 1 ) , then we have 2 g = ( 1 , 1 ) + ( 1 , 1 ) = ( 0 , 2 ) 3 g = 2 g + ( 1 , 1 ) = ( 0 , 2 ) + ( 1 , 1 ) = ( 1 , 0 ) 4 g = 3 g + ( 1 , 1 ) = ( 1 , 0 ) + ( 1 , 1 ) = ( 0 , 1 ) 5 g = 4 g + ( 1 , 1 ) = ( 0 , 1 ) + ( 1 , 1 ) = ( 1 , 2 ) 6 g = 5 g + ( 1 , 1 ) = ( 1 , 2 ) + ( 1 , 1 ) = ( 0 , 0 ) . In other words, Z 2 × Z 3 is cyclic of order 6, and hence isomorphic to Z 6 . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Example Consider Z 2 × Z 3 . It has 6 elements, namely ( 0 , 0 ) , ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 0 ) , ( 1 , 1 ) , and ( 1 , 2 ) . Since it is abelian, it must be isomorphic to Z 6 . (A group of order 6 is isomorphic to either Z 6 or S 3 .) In fact, if we set g = ( 1 , 1 ) , then we have 2 g = ( 1 , 1 ) + ( 1 , 1 ) = ( 0 , 2 ) 3 g = 2 g + ( 1 , 1 ) = ( 0 , 2 ) + ( 1 , 1 ) = ( 1 , 0 ) 4 g = 3 g + ( 1 , 1 ) = ( 1 , 0 ) + ( 1 , 1 ) = ( 0 , 1 ) 5 g = 4 g + ( 1 , 1 ) = ( 0 , 1 ) + ( 1 , 1 ) = ( 1 , 2 ) 6 g = 5 g + ( 1 , 1 ) = ( 1 , 2 ) + ( 1 , 1 ) = ( 0 , 0 ) . In other words, Z 2 × Z 3 is cyclic of order 6, and hence isomorphic to Z 6 . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Example Consider Z 2 × Z 2 . It has 4 elements. Thus, it is isomorphic to either Z 4 or Z × 8 = { 1 , 3 , 5 , 7 } . Now for all ( a , b ) ∈ Z 2 × Z 2 , we have 2 ( a , b ) = ( 0 , 0 ) . Thus, there is no element having order 4. The group must be isomorphic to Z × 8 . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
Direct products Finitely generated abelian groups Example Consider Z 2 × Z 2 . It has 4 elements. Thus, it is isomorphic to either Z 4 or Z × 8 = { 1 , 3 , 5 , 7 } . Now for all ( a , b ) ∈ Z 2 × Z 2 , we have 2 ( a , b ) = ( 0 , 0 ) . Thus, there is no element having order 4. The group must be isomorphic to Z × 8 . Instructor: Yifan Yang Section 11 – Direct products and finitely generated abelian groups
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