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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?

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  • $\begingroup$ No comment. Your thoughts have solved your own problem. $\endgroup$ – user83310 Sep 1 '15 at 5:20
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    $\begingroup$ @Jin Shin Great, thanks for taking the time to look it over! $\endgroup$ – user19817 Sep 1 '15 at 5:49

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