I want to proof the following equivalence: All units of a commutative group ring $\mathbb{Z}G$ are trivial $\Leftrightarrow$ for every $x \in G$ and every natural number $j$, relatively prime to $|G|$, we have $x^{j} = x$ or $x^{j} = x^{-1}$.
I already proved the '=>' part by using following theorem: " all units of $\mathbb{Z}G$ are trivial if and only if: (a) $G$ is abelian with exponent a divisor of 4 or 6 (b) $G =Q_8 \times E$, with $E$ an elementary abelian $2$-group. "
Can someone help me to prove the '<=' part please? I think I will also have to use the theorem above.
Thanks a lot, Daphné