Chapter5 - Chapter 5: Direct and Semidirect Products and...

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Unformatted text preview: Chapter 5: Direct and Semidirect Products and Abelian Groups Keith E. Emmert Direct Products The Fundamental Theorem of Finitely Generated Abelian Groups Table of Groups of Small Order Chapter 5: Direct and Semidirect Products and Abelian Groups Keith E. Emmert Tarleton State University March 5, 2010 Chapter 5: Direct and Semidirect Products and Abelian Groups Keith E. Emmert Direct Products The Fundamental Theorem of Finitely Generated Abelian Groups Table of Groups of Small Order Overview Direct Products The Fundamental Theorem of Finitely Generated Abelian Groups Table of Groups of Small Order Chapter 5: Direct and Semidirect Products and Abelian Groups Keith E. Emmert Direct Products The Fundamental Theorem of Finitely Generated Abelian Groups Table of Groups of Small Order Definition Definition 1 1. The direct product G 1 × G 2 × ··· × G n of the groups G 1 , G 2 , . . . , G n with operations ⊙ 1 , . . . , ⊙ n , is the set of n-tuples ( g 1 , . . . , g n ) where g i ∈ G i with operation defined component wise: ( g 1 , . . . , g n ) · ( h 1 , . . . , h n ) = ( g 1 ⊙ 1 h 1 , . . . g n ⊙ n h n ) . 2. The direct product G 1 × G 2 × ··· of the groups G 1 , G 2 , . . . with operations ⊙ 1 , ⊙ 2 , . . . , is the set of sequences ( g 1 , g 2 , . . . ) where g i ∈ G i with operation defined component wise: ( g 1 , g 2 , . . . ) · ( h 1 , h 2 , . . . ) = ( g 1 ⊙ 1 h 1 , g 2 ⊙ 2 h 2 , . . . ) . Chapter 5: Direct and Semidirect Products and Abelian Groups Keith E. Emmert Direct Products The Fundamental Theorem of Finitely Generated Abelian Groups Table of Groups of Small Order How Big Is Big? Proposition 2 If G 1 , . . . , G n are groups, their direct product is a group of order | G 1 || G 2 |···| G n | . Of course, if one of the groups is infinite, then the direct product has infinite order. Chapter 5: Direct and Semidirect Products and Abelian Groups Keith E. Emmert Direct Products The Fundamental Theorem of...
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This note was uploaded on 01/16/2012 for the course MATH 508 taught by Professor Staff during the Spring '08 term at Tarleton.

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Chapter5 - Chapter 5: Direct and Semidirect Products and...

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