I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group.

Is it true? Where can I find a proof?

A counterexample for open subsets of $\mathbb{R}^n$ is furnished by the halfplane with the $ax+b$ law.

  • 1
    $\begingroup$ Torsten, make it an answer (not just a comment). $\endgroup$ Apr 17, 2010 at 19:13
  • $\begingroup$ Moved a comment to an answer as per instructions. $\endgroup$ Apr 17, 2010 at 20:36

1 Answer 1


This is true and is in "Michel Lazard: Sur la nilpotence de certains groupes algébriques, Comptes Rendus, vol 241, 1955, 1687--1689"

  • 2
    $\begingroup$ This short paper is apparently not available online, but a version of Lazard's theorem is also written down in the 1970 book by Demazure-Gabriel, Groupes algebriques, I: see IV, section 4, 4.1. $\endgroup$ Apr 17, 2010 at 20:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.