Prove that An is simple for n ≥ 5, following the steps and hints given.a. Show A„ contains...

Prove that An is simple for n ≥ 5, following the steps and hints given.

a. Show A„ contains every 3-cycle if n ≥ 3.

b. Show A„ is generated by the 3-cycles for n ≥ 3. [Hint: Note that (a, b)(c\ d) = (a. c. b)(a. c, d) and (a,c)(a,fc) = (a, b, c).]

c. Let r and s be fixed elements of {1. 2, ....,n} for n ≥ 3. Show that A„ is generated by the n "special" 3-cycles of the form (r, s, i) for I ≤ i ≤ n [Hint: Show every 3-cycle is the product of "special" 3-cycles by computing (r, s, i)2, (r, s, j)(r, s, i)2, (r, s, j)2(r, s, i). and (r, s, i)2(r, s, k)(r, s, j)2(r, s, i).

Observe that these products give all possible types of 3-cycles.]

d. Let N be a normal subgroup of An for n ≥ 3. Show that if N contains a 3-cycle, then N = An. [Hint: Show that (r, s. i) ϵ N implies that (r. $, j) ϵ N for j = 1, 2..., n by computing ((r, s)(i, j))(r, s, i)2 ((r, s)(i, j))-1.]

e. Let N be a nontrivial normal subgroup of An for n ≥ 5. Show that one of the following cases must hold, and conclude in each case that N = An.

Case I N contains a 3-cycle.

Case II N contains a product of disjoint cycles, at least one of which has length greater than 3. [Hint: Suppose N contains the disjoint product σ μ(a1,a2, ..., ar). Show σ-1{{a1, a2, a3)σ{a1, a2, a3)-1 is in N, and compute it.]

Case III N contains a disjoint product of the form σ = μ(a4, a5, a6)(a1, a2, a3)- [Hint: Show σ-1 (a1, a2, a4) σ{a1, a2, a4)-1] is in N, and compute it.]

Case IV N contains a disjoint product of the form σ = μ(a1, a2, a3) where μ is a product of disjoint 2-cycles. [Hint: Show σ-2 ϵ N and compute it.]

Case V N contains a disjoint product a of the form σ = μ(a3, a4)(a1, a2), where μ is a product of an even number of disjoint 2-cycles. [Hint: Show that a-1(a1, a2, a3)σ{a1, a2, a3)-1 is in N, and compute it to deduce that α = (a2, a4)(a1, a3) is in N. Using N ≥ 5 for the first time, find i ≠a1, a2, a3, a4 in {1, 2, ...,n}. Let β = (a1, a3, i). Show that β-1αβα ϵ N, and compute it]