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aa_hw_6_Math 113

# 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 ) Ä ( G/K ). We define the map π : G/H → G/K by gH Ô→ gK . Suppose we pick two g,g Í ∈ G such that gH = g Í H . Then there is a h ∈ H s.t. g = g Í h . Note that all elements of H are also in K , since H < K by our assumption, so it also holds that gK = g Í K . This means that π ( gH ) = π ( g Í H ), so π is well defined. The kernel of the homomorphism are those gH ∈ G/K , s.t. π ( gH ) = K : ker( π ) = { gH ∈ G/H : π ( gH ) = K } = { gH ∈ G/H : gK = K } = { gH ∈ G/K : g ∈ K } = G/K. Since ker( π ) = G/K , we have by the first isomorphism theorem, that ( G/H ) / ( K/H ) Ä ( G/K ) and also ker( π ) G/H ⇔ K/H G/H . Problem 2 Let G be a group and N G . Let π : G → G/N be the canonical projection with π ( g ) = gN , and let H < G/N . We wish to show that π- 1 ( H ) < G ....
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aa_hw_6_Math 113 - Introduction to Abstract Algebra...

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