Let f:G->H be a group homomorphism.

Prove or disprove the following statement.

1.Let a be an element of G. If f(a) is of finite order, then a is also of finite order.

2.Let f be a surjection. Then f is an isomorphism iff the order of the element f(a) is equal to the order of the element a , for all a belong to G.

First, I want to know if f is the trivial homorphism, then both will fail, right?

Second, if f is non-trivial homomorphism. Both will hold, right?

Finally, Please help me to prove both or give me counterexamples without considering the trivial homomorphism.