Problems related to group homomorphism with kernel

Question

Problems:

• Let ƒ:G→ H  be a ^{group homomorphism with kernel} K = Ker(ƒ) , show that f is one to one if and only if K = {e_G}.

• Let K be a fixed integer and let be a function from a group G to itself with Ø(x) = x^k for all x∈G, show that Ø is a group homomorphism when G is abelian. Show by example that need not be a homomorphism when G is not commutative.

• Let Ø: G→ H and φ:H →K be two group homomorphism. Prove that φ⁰Ø :G->K is also a group homomorphism.

• Let G be the subgroup of the permutation group generated A6 by the permutation

Prove that G has order 6. Let be a group homomorphism from G to the permutation group which is also of order 6. By considering the image of , prove that cannot be an n isomorphism.

Prove that G has order 6. Let Ø be a group homomorphism from G to the permutation group which is also of order 6. By considering the image of σ , prove that Ø cannot be an n isomorphism.