yes if grp is abelian then we can say that group of order n is isomorphic when gcd of n and order of given mapping are relatively prime with each other
we know Group of prime order is cyclic
therefore G is cyclic and x⁷=e
f(x)=x⁴
f(x²)=x⁸=x
f(x³)=x¹²=x⁵
f(x⁴)=x¹⁶=x²
f(x⁵)=x²⁰=x⁶
f(x⁶)=x²⁴=x³
f(x⁷)=x²⁸=x⁷
so one one onto homom...