Originally Posted by

**tonio** I don't follow your hints: if $\displaystyle G=S_3$ , say and $\displaystyle n=3\,,\,\,g=(1)$ then there is NO unique solution to the equation $\displaystyle x^3=g$ in S_3....so why do you hint to take $\displaystyle G$ to be any finite group? Unless you meant that ANY finite group will fail the test...but then this hardly answers the OP's question, doesn't it?

I, for one, cannot think of any group at all, abelian or not, finite or not, that the required condition is true...

Tonio