Mth 461

HW 1

January 29, 2018

Due Monday, Feb 5. Be sure to justify all of your answers to receive full credit.

1. We consider two permutations, σ1,σ2 in S6.

σ1 :

 

1 → 5 2 → 3 3 → 6 4 → 1 5 → 4 6 → 2

σ2 :

 

1 → 4 2 → 6 3 → 1 4 → 2 5 → 5 6 → 3

(a) Compute the cycle decompositions of both σ1 and σ2.

(b) Compute σ1 ◦σ2,σ2 ◦σ1 and σ51. (c) What is the smallest positive integer k1 such that σ

k1 1 = e?

2. Recall that D8 is the group of symmetries of the square. Number the vertices 1, 2, 3, 4 counterclockwise, and let σ be rotation counterclockwise by 90 degrees, and τ reflection across the line y = x, which forms the diagonal connecting 1 and 3.

12

3 4

(a) Show that σ4, τ2, and σ ◦ τ ◦σ ◦ τ are all equal to the identity. (b) Show that every element of D8 has the form σ

k ◦τj, where 0 ≤ k ≤ 3 and 0 ≤ τ ≤ 1.

3. In this question we will investigate the sign of a permutation σ ∈ Sn, we we denote sgn(σ): Let c(σ) be the number of cycles in the cycle decomposition of σ, including fixed points. Then sgn(σ) is (−1)n−c(σ). For example, if σ ∈ S7 has the decomposition (1374)(2)(56), then n = 7, and c = 3, so sgn(σ) = 1.

(a) How many elements of S3 have sgn = 1? Of S4?

1

(b) Show that if σ is any permutation in Sn, and τ is a transposition in Sn (that is, a cycle of length 2), then either c(τ ◦σ) = c(σ) + 1 or c(τ ◦σ) = c(σ) − 1. Thus

sgn(τ ◦σ) = −sgn(σ).

(c) Use induction on n to prove that the product of n transpositions has sgn(−1)n. Conclude that for any σ1,σ2 ∈ Sn,

sgn(σ1 ◦σ2) = sgn(σ1)sgn(σ2).

4. Saracino: 0.5, 0.9, 1.3.

2