A note on the Lie polynomial $ \mbf{(x+y)^{p}-x^{p}-y^{p}} $

Abstract

Using the techniques of group action on a set, we will give an elementary ane complete proof in this paper that the Lie polynomial $ (x+y)^{p}-x^{p}-y^{p} $ over prime field $ F_{p} $ is a sum of Lie brackets, with $ p $ a rational prime.