Lazard filtration on a group algebra

For a p-valued group $(G,\omega )$, give $p^r(g_1-1)\cdots (g_s-1)$ weight $r+{\sum }_i\omega (g_i)$. The spans of elements of weight at least $\lambda$ form a multiplicative filtration on ${\mathbb{Z}}_p[G]$ and its completions.

The initial form of $g-1$ depends only on the initial form of $g$ in $\mathrm{gr}G$. The resulting Lie map extends to a surjective graded algebra homomorphismFor a complete finite-rank p-valued group, ordered-basis expansions show that it is an isomorphism.