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

Abstract

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.