Exercise 14, § 7

A group G is said to be free if it possesses a basis, and hence is isomorphic to a free group F(X) constructed on a set X. (a) Prove that, if $(t_i)_{i \in I}$ and $(t'_j)_{j \in J}$ are two such families, one has Card(I) = Card(J). (When Card(I) is infinite, one will prove that the same is true of Card(J), and that both are equal to Card(G). When Card(I) is an integer d, one will prove that the number of subgroups of index 2 of G is $2^d - 1$.) The cardinal of a basis of G is called the rank of G. (b) A free group is of rank 0 (resp. 1) if and only if it is reduced to $e$ (resp. isomorphic to $\mathbf{Z}$). (c) Let G be a free group of finite rank d, and let $(x_1, \ldots, x_d)$ be a generating family with d elements of G. Prove that this family is a basis. (Use I, p. 149, exerc. 34, and I, p. 129, exerc. 5.) (d) Prove that a free group of rank $\geq 2$ contains a free subgroup of given rank d, for every cardinal $d \leq \aleph_0$. (Use I, p. 147, exerc. 22.)