Exercise 3(20 points) a) Let : G→ H be a group homomorphism. Let g € G be an element of finite order. Prove that (g) has

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Exercise 3 20 Points A Let G H Be A Group Homomorphism Let G G Be An Element Of Finite Order Prove That G Has 1 (136.22 KiB) Viewed 37 times

Exercise 3(20 points) a) Let : G→ H be a group homomorphism. Let g € G be an element of finite order. Prove that (g) has finite order in H, 1 and show that the order of (g) divides the order of 9. Now let : G→ H is an isomorphism, and let g € G. We know is a homomorphism implies ord(o(g)) divides ord(g). = g divides Also ¹ is a homomorphism implies ord(¯¹(0(g))) ord(p(g)). This implies if is an isomorphism then ord(p(g)) = ord(g). for all g € G. b) Consider the two groups and G = {< x, y > |x¹ = 1; y² = 1; xy³ = yx}. G' =< x, y > |xª¹ = 1; y² = x²; x³y = yx Prove that G is not isomorphic to G'.