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) quotient out by the kernel, and then (ii) relabel the nodes via . Let G be a group and let G be the commutator subgroup of G. Prove that the quotient group G/G is abelian and that G' is the smallest normal subgroup of G for which the quotient is abelian. Visualizing the FHT via Cayley graphs G (Ker(6) G) Im(6) H any homomorphism iN quotient process G/Ker(d) group of cosets remaining isomorphism ("relabeling") Qg iN kN "quotient map" N $ = LOTT QB/N jN iN kN VA "relabeling map"