Exercise 14, § 7

A group $G$ is called free if it possesses a basic family and hence is isomorphic to a free group $F(X)$ constructed on a set $X$. (a) Show that, if $(t_i)_{i \in I}$ and $(t'_j)_{j \in J}$ are two such families, then $\mathrm{Card}(I) = \mathrm{Card}(J)$. (When $\mathrm{Card}(I)$ is infinite, show that $\mathrm{Card}(J)$ is also and that both are equal to $\mathrm{Card}(G)$. When $\mathrm{Card}(I)$ is an integer $d$, show that the number of subgroups of index 2 in $G$ is $2^d - 1$.) The cardinal of a basic family of $G$ is called the rank of $G$. (b) A free group is of rank 0 (resp. 1) if and only if it reduces to $\{e\}$ (resp. is isomorphic to $\mathbf{Z}$). (c) Let $G$ be a free group of finite rank $d$ and $(x_1, \ldots, x_d)$ a generating family of $d$ elements of $G$. Show that this family is basic. (Use Exercise 34 and § 5, Exercise 5.) (d) Show that a free group of rank $\geq 2$ contains a free subgroup of given rank $d$ for every cardinal $d \leq \aleph_0$. (Use Exercise 22.)