《Discrete Mathematics II教学-华南理工》Section 11 Direct Products and Finitely Generated Abelian Groups.pdfVIP

II.11 Direct Products, Finitely Generated Abelian Groups 1 Section II.11. Direct Products and Finitely Generated Abelian Groups Note. In the previous section, we took given groups and explored the existence of subgroups. In this section, we introduce a process to build new (bigger) groups from known groups. This process will allow us to classify all finite abelian groups. Definition 11.1. The Cartesian product of sets S , S , . . . , S of the set of all 1 2 n ordered n-tuples (a , a , . . . , a ) where a ∈ S for i = 1, 2, . . . , n. This is denoted 1 2 n i i n S = S × S × · · · × S . i 1 2 n i=1 Theorem 11.2. Let G , G , . . . , G be (multiplicative) groups. For (a , a , . . . , a ), 1 2 n 1 2 n (b , b , . . . , b ) ∈ G , define the (multiplicative) binary operation 1 2 n i (a , a , . . . , a )(b , b , . . . , b ) = (a b , a b , . . . , a b ). 1 2 n 1 2 n 1 1 2 2 n n The Gi is a group under this binary operation, called the direct product of the groups G . i II.11 Direct Products, Finitely Generated Abelian Groups 2 Note. If each Gi is an additive group, then we may refer to Gi as the direct sum of the groups Gi and denote it as G