Prove that 𝐴􀯡 is non-abelian for 𝑛 ≥ 4.

The Alternating Group A_n is a group of even permutations of a finite set. A group is said to be non-abelian if it does not follow the commutative property, i.e., the order of the elements matters in the operation of the group.

To prove that A_n is non-abelian for n ≥ 4, we can use the fact that a permutation group is non-abelian if it contains a permutation of order 3 and a permutation of order 2 that are not disjoint.

Step 1: Consider the set {1, 2, 3, 4} for n=4.

Step 2: Let's take two permutations: σ = (123) and τ = (34). Here, σ is a permutation of order 3 and τ is a permutation of order 2.

Step 3: We can see that σ and τ are not disjoint as they both involve the element 3.

Step 4: Now, let's compute στ and τσ:

στ = (123)(34) = (124) τσ = (34)(123) = (134)

Step 5: Since στ ≠ τσ, we can conclude that the group A_n is non-abelian for n ≥ 4.

This proof can be generalized for all n ≥ 4 by considering the set {1, 2, ..., n} and choosing appropriate permutations of order 2 and 3 that are not disjoint.