Please do not just copy paste from AI, I need original work.Fundamental homomorphism theorem (FHT)If 6: G H is a homomorphism, then Im($) G/Ker().The FHT says that every homomorphism can be decomposed into two steps: (i) quotientout by the kernel, and then (ii) relabel the nodes via p.Let G be a cyclic group of order n, ie, G = (g), and let H be a normal subgroup of G oforder m, where m divides n. Show that the quotient group G/H is cyclic, and determine itsorder.Visualizing the FHT via Cayley graphsG(Ker(6) G)Im(6) HQ8any homomorphismiNquotientprocessG/Kerid)group ofcosetsremaining isomorphism("relabeling")iNkN"quotient map" πTNjN$ = LOTTQB/NiNkNVA"relabeling map"

Approach to solving the question:Showing that G/H, the quotient group, is cyclic and determining the…

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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 32E: 32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping ...

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