If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows

$$\mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \gamma_{k+1}(G)$$

with the structure of a Lie ring. A famous theorem that should probably be attributed to Magnus says that if $G$ is the free group on $n$ generators, then $\mathcal{L}(G)$ is the free Lie ring on $n$ generators. This result is documented in many places, e.g. Magnus--Karass--Solitar's book on combinatorial group theory and Serre's book Lie Group and Lie Algebras.

Now fix a prime $p$ and let $\delta_k(G)$ be the fastest descending central series of $G$ satisfying the following three properties:

  1. $\delta_1(G) = G$, and

  2. $[\delta_k(G),\delta_{\ell}(G)] \subset \delta_{k+\ell}(G)$ for all $k,\ell \geq 1$, and

  3. $(\delta_k(G))^p \subset \delta_{pk}(G)$ for all $k \geq 1$.

This series was first defined by Zassenhaus. The commutator bracket and the $p$th power operation on $G$ endow

$$\Lambda(G) = \bigoplus_{k=1}^{\infty} \delta_k(G) / \delta_{k+1}(G)$$

with the structure of a restricted Lie algebra over the field $\mathbb{F}_p$.

I'm pretty certain that I can prove that if $G$ is a free group on $n$ generators, then $\Lambda(G)$ is the free restricted Lie algebra on $n$ generators over the field $\mathbb{F}_p$. I need this result for a paper I am writing (on an unrelated topic), but I'm certain that this result is known and would greatly prefer to just cite a reference for it.

Question: Can anyone give me a reference for this?


This is Theorem 6.5 p.130 from Lazard, Michel. Sur les groupes nilpotent et les anneaux de Lie. Ann. Sci. Ecole Norm. Sup (3) 71 (1954) 101-190.


I cannot access the following paper: Baumann, Bernd(D-GSSN); Scoppola, Carlo M.(I-TRNT) Canonical bases for normal subgroups of finitely generated free groups with finite abelian factor groups. (English summary) Forum Math. 5 (1993), no. 1, 23–32. However, I think it might contain an answer to your question.


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