# Trivial units of commutative group ring

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

• What does it mean for a unit to be "trivial"? – Randall Oct 18 '17 at 13:08
• @Randall In the context of group algebras, the trivial units of $R[G]$ are the elements of $\{ug\mid u\in R^\times, g\in G\}$ – rschwieb Oct 18 '17 at 13:17
• Dear @daphne Generally it's better not to clutter your post with things like signatures and salutations. No big deal this time, but just try to leave them out in the future. Thanks – rschwieb Oct 18 '17 at 13:19