Exercise 9, § 5

Let $X$ be the set with two elements $\{x, y\}$.

(a) Show that there exists a sequence $w_2, w_3, \ldots$ of elements of $\mathbf{F}$ such that $w_i \in \mathrm{C}^i\mathbf{F}$ for all $i$ and

† For more details on typical sequences, see M. Lazard, Annales E. N. S., 71 (1954), pp. 101–190, Chapter II, §§ 1, 2.

for all $n \in \mathbf{Z}$ and all $i \geqslant 1$. (Apply Exercise 8 (d) to the sequence of $x^{-n}(xy)^n$.) Show that $w_2 = y^{-1}(y^{-1}, x^{-1})y \equiv (x, y)^{-1} \mod C^3F$. (b) Let G be a nilpotent group of class $\leq c$ and let $x, y$ be in G. Deduce from the above the following formula ("Hall's formula");

(c) Let $p$ be a prime number. Show that there exists $u_i \in C^iF$ ($2 \leq i \leq p - 1$) and $w \in C^pF$ such that

(Use (a), noting that $\binom{p}{i}$ is divisible by $p$ if $1 < i < p$.) Let $\bar{w}$ be the image of $w$ in $\mathrm{gr}_p(F) = L_p^p(X)$ and let $\bar{w}_p$ be the image of $\bar{w}$ in $L_{F_p}^p(X)$. Show that $\bar{w}_p = \Lambda_p(x, y)$, cf. Chapter I, § 1, Exercise 19. (Use the extension $g$ of F to $\hat{\Lambda}(X)$ and compare the terms of degree $p$ in $g((xy)^p)$ and $g(x^p y^p u_2^p \ldots u_{p-1}^p w)$. The first is equal to $(x + y)^p$ and the second is congruent mod. $p$ to $x^p + y^p + \bar{w}_p$. Hence the result.) Show that there exist $a \in C^2F$ and $w' \in C^pF$ such that (Use the above formula for $(xy)^p$.) Deduce that in a nilpotent group of class $p$ the $p$-th powers form a subgroup. (d) Show that there exists a sequence $v_3, v_4, \ldots$ of elements of F, where $v_i \in C^iF$ for all $i$ and for all $n \in \mathbf{N}$ and all $i \geq 1$. (Apply Exercise 8 (f) to the sequence of $(x^n, y)$.) Deduce that, if $p$ is a prime number, there exists $t_i \in C^iF$ for $3 \leq i \leq p$ and $z \in C^{p+1}F$ such that Show that the image $\bar{z}_p$ of $z$ in $L_{F_p}^{p+1}(X)$ is $(\mathrm{ad}\, x)^p(y)$. (Same method as for (c).)