Please do not just copy paste from AI, I need original work.Fundamental homomorphism theorem (FHT)If : G→ H is a homomorphism, then Im(#) G/Ker().The FHT says that every homomorphism can be decomposed into two steps: (i) quotientout by the kernel, and then (ii) relabel the nodes via .Let G be a finite group and N a normal subgroup of G. Prove that the order of the quotientgroup G/N is given by the index of N in G, i.e.,G|G/N|=NVisualizing the FHT via Cayley graphsGIm(0) H(Ker()G)any homomorphismiNquotientprocessG/Ker()remaining isomorphism("relabeling")Q8"quotient map"πiNkNN= LOTQo/NiNVA"relabeling map"

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 11E: Find all homomorphic images of the quaternion group.

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Find all subgroups of the quaternion group.
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Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
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