Theorem 6.25: Let op be a homomorphism function from group G; to group G; Then g/Ker(φ);|φ(G); e isomorphism function f: G/ Ker(o)-(p(G) Let K= Ker(). For VKaEG/K, f(Ka=op(a) Is an isomorphism iunction ◆ Proof: For v Ka∈G/kK,letf(Ka)=q(a) (fis an everywhere function from G/K to p(G) For Ka=kb, cp(a)=?p(b) +(2) is a homomorphism function ◆ For v Ka,kb∈G/K,f(Kakb)=?fka)efkb) ◆(3) fis a bijection ◆One-to-one ◆Onto Theorem 6.25：Let  be a homomorphism function from group [G;*] to group [G';•]. Then [G/Ker();][(G);•]  isomorphism function f:G/ Ker()→(G).  Let K= Ker(). For KaG/K，f(Ka)=(a)  f is an isomorphism function。  Proof: For  KaG/K，let f(Ka)=(a)  (1)f is an everywhere function from G/K to (G)  For Ka=Kb,(a)=?(b)  (2)f is a homomorphism function  For  Ka,KbG/K, f(KaKb)=?f(Ka)•f(Kb)  (3) f is a bijection  One-to-one  Onto