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Unformatted text preview: Chapter 3: Quotient Groups and Homomorphisms Keith E. Emmert Definition and Examples More on Cosets and Lagranges Theorem The Isomorphism Theorems Composition Series and the Holder Program Chapter 3: Quotient Groups and Homomorphisms Keith E. Emmert Tarleton State University March 4, 2010 Chapter 3: Quotient Groups and Homomorphisms Keith E. Emmert Definition and Examples More on Cosets and Lagranges Theorem The Isomorphism Theorems Composition Series and the Holder Program Overview Definition and Examples More on Cosets and Lagranges Theorem The Isomorphism Theorems Composition Series and the Holder Program Chapter 3: Quotient Groups and Homomorphisms Keith E. Emmert Definition and Examples More on Cosets and Lagranges Theorem The Isomorphism Theorems Composition Series and the Holder Program Fibers (no, not the cotton kind) Definition 1 Let : G H be a homomorphism and suppose h H . Then the fiber over h is defined to be X h = { g G  ( g ) = h } =  1 ( h ) . Remark 2 Let : G H be a homomorphism. 1. Let h 1 , h 2 H. Then X h 1 = X h 2 h 1 = h 2 . 2. G = uniondisplay h H X h . Thus, the group G is partitioned by the fibers over elements of H. 3. Let h 1 , h 2 H. Then we define a natural product on the fibers by X h 1 X h 2 = X h 1 h 2 . 4. Using the above product, the set of fibers forms a group. Chapter 3: Quotient Groups and Homomorphisms Keith E. Emmert Definition and Examples More on Cosets and Lagranges Theorem The Isomorphism Theorems Composition Series and the Holder Program Example Example 3 Consider the additive group G = Z . Let H = Z n = ( x ) , the cyclic group of order n . Define ( a ) = x a . Find the fiber of over x a . Chapter 3: Quotient Groups and Homomorphisms Keith E. Emmert Definition and Examples More on Cosets and Lagranges Theorem The Isomorphism Theorems Composition Series and the Holder Program Basic Ideas Definition 4 If is a homomorphism : G H , the kernel of is the set Ker ( ) = { g G  ( g ) = 1 H } . Proposition 5 Let G and H be groups and let : G H be a homomorphism. 1. ( 1 G ) = 1 H . 2. ( g 1 ) = ( g ) 1 for all g G. 3. ( g n ) = ( g ) n for all n Z . 4. Ker ( ) is a subgroup of G. 5. Im ( ) , the image of G under , is a subgroup of H. Chapter 3: Quotient Groups and Homomorphisms Keith E. Emmert Definition and Examples More on Cosets and Lagranges Theorem The Isomorphism Theorems Composition Series and the Holder Program Quotient Groups Definition 6 Let : G H be a homomorphism with kernel K . The quotient group or factor group , G / K , read G modulo K or G mod K , is the group whose elements are the fibers of with group operation defined by: X a X b = X ab for elements a , b H . Chapter 3: Quotient Groups and Homomorphisms Keith E. Emmert Definition and Examples More on Cosets and Lagranges Theorem The Isomorphism Theorems Composition Series and the Holder Program...
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