A partial extension of Lazard's correspondence for finite $p$-groups

Abstract

M. Lazard established a correspondence between finite $p$-groups of nilpotence class less than p and finite nilpotent Lie rings of $p$-power order and nilpotence class less than $p$. This correspondence has had many applications, but cannot generally be extended to $p$-groups of class $p$ or larger. However, in this paper, we obtain a partial extension of Lazard's result for a $p$-group that is a product of normal subgroups of class less than $p$.