HW6_Math 113

# HW6_Math 113 - MATH 113 HW#6 KEVIN JORGENSEN 1 Prove the...

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Unformatted text preview: MATH 113 HW #6 KEVIN JORGENSEN 1) Prove the Third Isomorphism Theorem: Let G be a group and let H and K be normal subgroups of G such that H is a subgroup of K . Then K/H is a normal subgroup of G/H and ( G/H ) / ( K/H ) is isomorphic to G/K . Suppose H,K E G and H ≤ K . It is trivial to see that K/H is a subgroup of G/H . Let gH ∈ G/H and kH ∈ K/H . Then gH ◦ kH ◦ ( gH )- 1 = gH ◦ kH ◦ g- 1 H = ( gk ) H ◦ g- 1 H = ( gkg- 1 ) H ∈ G/H. So K/H is a normal subgroup of G/H . Let ϕ : G/H → G/K , where gH 7→ gK . (Since H ≤ K , this is well-defined. Further, this map is clearly a surjection, as the choice of g is arbitrary.) This mapping is also very clearly a homomorphism: ϕ ( aH ◦ bH ) = ϕ (( ab ) H ) = ( ab ) K . ϕ ( aH ) ◦ ϕ ( bH ) = aK ◦ bK = ( ab ) K . Finally, ker ( ϕ ) = { gH ∈ G/H | ϕ ( gH ) = K } = { gH ∈ G/H | gK = K } = { gH ∈ G/H | g ∈ K } = K/H And by the First Isomorphism Theorem, ( G/H ) /ker ( ϕ ) ∼ = ϕ ( G/H ), so ( G/H ) / ( K/H ) ∼ = G/K , since ϕ is surjective. Q.E.D. 1 2 KEVIN JORGENSEN 2) Let G be a group and let N be a normal subgroup. Let π N : G → G/N be the canonical projection and let H < G/N . Show that π- 1 ( H ) is a subgroup of 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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HW6_Math 113 - MATH 113 HW#6 KEVIN JORGENSEN 1 Prove the...

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