Question

Show that if n≥3 then every element in Aₙ is a product of 3 -cycles. Note: By definition, if a permutation σ belongs to Aₙ, then σ can be written as the product of an even number of transpositions. Hint: If you look at the composition of two transpositions, there are three possibilities:

(a) (ij)(kℓ) with i,j,k,ℓ distinct,
(b) (ij)(iℓ) with i,j,ℓ distinct,
(c) (ij)(ij) with i,j distinct.

Answer 1

To show that every element in Aₙ is a product of 3-cycles, we can demonstrate that every element in Aₙ can be written as the product of an even number (or zero) of 2-cycles or transpositions. Using the given hint and considering the three possibilities of composing two transpositions, we can prove that each composition results in a 3-cycle.

Step-by-step explanation:

To show that every element in Aₙ is a product of 3-cycles, where n≥3, we need to demonstrate that every element in Aₙ can be written as the product of an even number (or zero) of 2-cycles or transpositions.

Given the hint, let's consider the composition of two transpositions:

1. (a) (ij)(kℓ) with i,j,k,ℓ distinct
2. (b) (ij)(iℓ) with i,j,ℓ distinct
3. (c) (ij)(ij) with i,j distinct

We can prove that each of these compositions results in a 3-cycle:

• (a) (ij)(kℓ) = (ikℓ)
• (b) (ij)(iℓ) = (jℓi)
• (c) (ij)(ij) = e, the identity permutation

Therefore, since every element in Aₙ can be written as the product of 2-cycles that result in 3-cycles, every element in Aₙ is indeed a product of 3-cycles.