Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups

We define a wreath product of a Lie algebra $L$ with the one-dimensional Lie algebra $L_1$ over $\mathbb F_p$ and determine some properties of this wreath product. We prove that the Lie algebra associated with the Sylow p-subgroup of finite symmetric group $S_{p^m}$ is isomorphic to the wreath product of $m$ copies of $L_1$. As a corollary we describe the Lie algebra associated with Sylow $p$-subgroup of any symmetric group in terms of wreath product of one-dimensional Lie algebras.