If f: Z -> Z is the map defined by f(x) = 2x. Is f a group homomorphism when the group operation on Z is addition? How would you prove it?

If f: Z→ Z is the map defined by f(x)=2x, Is f a ring homomorphism when Z has its usual ring operations? How would you prove that? If not, what could be a counter-example?

Consider the set of matrices 0 S = {(a+b):abER} a, bЄR

2* Let f G H be a group homomorphism. Prove that if x E G and n is a natural number then f(x)= f(x)"