Lazard logarithm of a p-saturated group

Inside the completed rational Iwasawa algebra, the convergent seriessends a finite-rank p-saturated group to a ${\mathbb{Z}}_p$-Lie algebra. The Baker--Campbell--Hausdorff formula and its commutator expansion prove closure under addition and the commutator bracket.

If $(g_1,\dots ,g_d)$ is an ordered basis of a finite-rank p-saturated group, thenwhere a primitive element $x$ satisfies $\Delta (x)=x\otimes 1+1\otimes x$ for the completed Hopf algebra coproduct.