Let $G$ be a finite group and $F$ a finite field with co-prime orders. Show that in the group algebra $F[G]$, if $x \in F[G]$ is not a left zero divisor then it is invertible.
Thoughts so far: By Maschke's Theorem $F[G]$ is a semi-simple algebra, hence isomorphic to a product of matrix rings over division rings, but the division rings are finite, hence fields. And a product of matrix rings over fields has the given property. Am I on the right track?