aa_hw_6_Math 113 - Introduction to Abstract Algebra...

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Unformatted text preview: Introduction to Abstract Algebra Homework 6 October 18, 2010 Problem 1 We prove the third isomorphism theorem, stating that if H,K G and H < K , then K/H G/H and ( G/K ) / ( K/H ) ( G/K ). We define the map : G/H G/K by gH gK . Suppose we pick two g,g G such that gH = g H . Then there is a h H s.t. g = g h . Note that all elements of H are also in K , since H < K by our assumption, so it also holds that gK = g K . This means that ( gH ) = ( g H ), so is well defined. The kernel of the homomorphism are those gH G/K , s.t. ( gH ) = K : ker( ) = { gH G/H : ( gH ) = K } = { gH G/H : gK = K } = { gH G/K : g K } = G/K. Since ker( ) = G/K , we have by the first isomorphism theorem, that ( G/H ) / ( K/H ) ( G/K ) and also ker( ) G/H K/H G/H . Problem 2 Let G be a group and N G . Let : G G/N be the canonical projection with ( g ) = gN , and let H < G/N . We wish to show that - 1 ( H ) < G ....
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This note was uploaded on 10/15/2011 for the course CHEM 1 taught by Professor Gelfand during the Summer '11 term at Solano Community College.

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aa_hw_6_Math 113 - Introduction to Abstract Algebra...

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