To show a mapping is isomorphic you must show 2 things (ok really 3 things), that the mapping is is a bijection and that is preserves operation.
To show that it is 1-1 assume that P(a)=P(b) (P is phi). Prove that a=b.
To show that it is onto you must show that for any element g* in G* (G* is the group that G maps to) there exists a g in G that gets mapped to it, i.e. P(g)=g* .
Finally show that the mapping preserves operation, that is P(ab)=P(a)P(b) for all a,b in G.