For (a), what have you done about the first part? For the second part, you need a counter-example. Clearly, the kernel of your mapping must be trivial (why?) and so your mapping must be 1-1. If it is an injective homomorphism but NOT an isomorphism, what then can you say?

For (b) you need to remember that your isomorphism is also a bijection, and so the two groups contain the same number of elements. Each element in the image has the same order as in the pre-image, and so they must tally.