Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If o: G+ H is a homomorphism, then Im(4)G/Ker(). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via . Let G be a group and H a subgroup of G. Show that there exists an exact sequence involving the quotient group G/H and its relation to homomorphisms. Specifically, prove that for a homomorphism : GK where K is a group, the induced map | H→ K can be extended to a homomorphism from G/H to K. Visualizing the FHT via Cayley graphs G Im(6) H (Ker(6) 4G) any homomorphism iN quotient process G/Ke group of cosets remaining isomorphism ("relabeling") Q8 iN KN "quotient map" π N jN = LOT QB/N iN kN VA "relabeling map"