Exercise 14, § 4

Let $P$ be a set of prime numbers. An integer $n$ is called a $P$-integer if it is $\neq 0$ and all its prime factors belong to $P$. An element $x \in G$ is called a $P$-torsion element if there exists a $P$-integer $n$ such that $x^n = e$; $G$ is called a $P$-torsion (resp. P-torsion free) group if every element of G is a P-torsion element (resp. if no element of G other than e is a P-torsion element). G is called P-divisible if, for all $x \in G$ and every P-integer $n$, there exists $y \in G$ such that $x = y^n$.

Suppose that G is nilpotent.

(a) Let H be a subgroup of G and let $H_P$ be the set of $x \in G$ such that there exists a P-integer $n$ for which $x^n \in H$. Show that $H_P$ is a subgroup of G (use Exercise 12) and that $(H_P)_P = H_P$. The group $H_P$ is called the P-saturation of H in G. If G is P-divisible, so is $H_P$.

(b) Let L be a normal subgroup of H. Show that $L_P$ is normal in $H_P$. (Use Exercise 13 (c) to prove that, if $g \in H_P$ and $l \in L$, then $glg^{-1} \in L_P$.)

In particular, if L is normal in G, so is $L_P$ and $G/L_P$ is P-torsion-free. Deduce that the set of P-torsion elements of G is the smallest normal subgroup N of G such that $G/N$ is P-torsion-free.

(c) Suppose that G is P-torsion-free. Let $n$ be a P-integer and let $x, y \in G$ be such that $x^n = y^n$. Show that $x = y$. (Apply Exercise 10 (a) with $r = s = n$. Deduce that there exists a P-integer N such that $(x, y)^N = e$, whence $(x, y) = e$ since G is P-torsion-free; then $(x^{-1}y)^n = e$, whence finally $x = y$.)

(d) Let H be a subgroup of G, let L be a nilpotent group and let $f : H \to L$ be a homomorphism. Let $\Gamma$ be the graph of $f$ in $H \times L$ and let $\Gamma_P$ be its P-saturation in $G \times L$. Show that $\Gamma_P$ is contained in $H_P \times L$ and that $\operatorname{pr}_1 : \Gamma_P \to H_P$ is surjective if L is P-divisible and injective if L is P-torsion-free.

Suppose that L is P-divisible and P-torsion-free. Show that $f$ can be extended uniquely to a homomorphism $f_P : H_P \to L$ and that the graph of $f_P$ is $\Gamma_P$.