Exercise 10, § 1

Let U_n be the alternating group consisting of the permutations w ∈ S_n whose signature is equal to +1. Show that U_n is the commutator subgroup of S_n (use Exerc. 4 and 9). For any integer i with 1 ≤ i ≤ n - 2, put u_i = s_i s_{i+1} (in the notation of Exerc. 4). Show that the family (u_i) and the relations u_1^3 = 1, u_i^2 = 1 for i ≥ 2, (u_i u_{i+1})^3 = 1 for 1 ≤ i ≤ n - 3, and u_i u_j = u_j u_i for 1 ≤ i ≤ n - 4 and i + 2 ≤ j ≤ n - 2, form a presentation of the group U_n (use Exerc. 9).