Exercise 6, § 5

Let $P$ be a set of prime numbers, let $S_P$ be the set of $P$-integers (§ 4, Exercise 14) and let $\mathbf{Z}_P = S_P^{-1}\mathbf{Z}$.

(a) Suppose that, for all $s \in S_P$, the mapping $k \mapsto sk$ is a bijection of $K$ onto itself; this is equivalent to saying that $K$ can be given a $\mathbf{Z}_P$-algebra structure.

With the notation of Exercise 5, show that, for all $s \in S_P$, the mapping $a \mapsto a^s$ of $\Gamma$ into itself is bijective and that the same is true for each quotient $\Gamma/\Gamma_n$ for $n \geq 1$. If $t \in \mathbf{Z}_P$ and $a \in \Gamma$ (resp. $a \in \Gamma/\Gamma_n$), $a^t$ is defined as in Exercise 16 of § 4; writing $a$ in the form $1 + \alpha$, where $\alpha \in \hat{A}_1(X)$, show that

(Note that the coefficients $\binom{t}{n}$ belong to $\mathbf{Z}_P$, cf. Exercise 4.)

(b) Suppose now that $K = \mathbf{Z}_P$. Let $c$ be an integer $\geq 1$. The group $F^c = F/C^cF$ is identified with a subgroup of $\Gamma/\Gamma_c$ by means of the homomorphism derived from g by taking quotients (notation of Theorem 2). The group $\Gamma/\Gamma_c$ is a P-torsion-free P-divisible group of class $<c$. Let $F_P^c$ be the P-saturation of $F^c$ in $\Gamma/\Gamma_c$ (§ 4, Exercise 14) and let i be the injection of $F^c$ into $F_P^c$. The ordered pair $(i, F_P^c)$ is a P-envelope of $F^c$ (§ 4, Exercise 15). Deduce that every nilpotent group has a P-envelope (note that every nilpotent group of class $<c$ is the quotient of a group $F^c$ for suitable X and use part (b) of Exercise 15 of § 4).

(c) If $n \leq c$, let $F_{P,n}^c$ be the intersection of $F_P^c$ with $\Gamma_n/\Gamma_c$; if $n \geq c$, we write $F_{P,n}^c = \{e\}$. Show that $F_{P,n}^c$ is the P-saturation of $C^n F^c$ in $\Gamma/\Gamma_c$. The filtration $(F_{P,n}^c)$ is an integral central filtration of $F_P^c$; let $\mathrm{gr}(F_P^c)$ be the graded module associated with this filtration. Show that, if $n < c$, the image of $\mathrm{gr}_n(F_P^c)$ in $\mathrm{gr}_n(\Gamma) = A_{\mathbf{Z}_p}(X)$ is $S_P^{-1}L_{\mathbf{Z}}(X) = L_{\mathbf{Z}_p}(X)$. Deduce that the Lie algebra $\mathrm{gr}(F_P^c)$ is generated by its elements of degree 1 and hence that $F_{P,n}^c = C^n(F_P^c)$ for all $n$ (§ 4, Exercise 1). Show that the group $F_P^c$ is generated by the $x^{1/s}$ for $x \in X, s \in S_P$ (note that the images of these elements in $\mathrm{gr}_1(F_P^c) \simeq \mathbf{Z}_p^{(X)}$ generate the group $\mathrm{gr}_1(F_P^c)$ and apply Corollary 3 to Proposition 8 of Algebra, Chapter I, § 6, no. 3).

(d) Let H be a Hall set relative to X (§ 2, no. 10) and let $H(c)$ be the subset of H consisting of the elements of length $<c$. For $m \in H(c)$, let $\phi_c(m)$ denote the image in $F^c$ of the basic commutator $\phi(m)$ defined by m (cf. no. 4, Remark). Show that, for all $w \in F_P^c$, there exists a unique element $\alpha \in \mathbf{Z}_P^{(H(c))}$ such that

(Use the determination of $\mathrm{gr}(F_P^c)$ obtained above.)