Exercise 16, § 4

We preserve the notation of the two previous exercises. (a) Let S_P be the set of P-integers and let $\mathbf{Z}_P = S_P^{-1}\mathbf{Z}$ be the ring of fractions of $\mathbf{Z}$ defined by S_P (Commutative Algebra, Chapter II, § 2, no. 1). Suppose that G is nilpotent, P-torsion-free and P-divisible and let $t \in \mathbf{Z}_P, g \in G$. Show that there exists one and only one element h of G such that $h^s = g^{st}$ for all $s \in S_P$ such that $st \in \mathbf{Z}$. The element h is called the t-th power of g and denoted by $g^t$. The mapping $t \mapsto g^t$ is a homomorphism of $\mathbf{Z}_P$ into G. If t is invertible in $\mathbf{Z}_P$, $g \mapsto g^t$ is bijective. (Use Exercise 14 (c)). (b) Let A be a commutative group and let i be the canonical mapping of A into $A_P = \mathbf{Z}_P \times A$. Show that $(i, A_P)$ is a P-envelope of A. (c) Let G (resp. $\overline{G}$) be the lower strict triangular group of order n over a ring k (resp. over the ring $k_P = k \otimes \mathbf{Z}_P$) and let i be the homomorphism of G into $\overline{G}$ defined by the canonical homomorphism of k into $k_P$ (Algebra, Chapter II, § 10, no. 4). Show that $(i, \overline{G})$ is a P-envelope of G. (d) Let G be a finite nilpotent group and hence the direct product of its Sylow p-groups $G_p$ (Algebra, Chapter I, § 6, no. 7, Theorem 4). Let $\overline{G}$ be the product of the $G_p$ for $p \notin P$ and let i be the canonical projection of G onto $\overline{G}$. Show that $(i, \overline{G})$ is a P-envelope of G.