Chapter 3 Definitions

Question 1

Cycle

Answer 1

A cycle in S_{n} is a permutation of district elements (i_{1},…i_{m}) such that

i_{1} → i_{2}
i_{2} → i_{3}
...
i_{m-1} → i_{m}
i_{m} → i_{1}

k → k for each k ∈ [n] \ (i_{1},…i_{m})

Question 2

Transposition

Answer 2

A length 2 cycle; (i_{1},i_{2})

Question 3

Lemma 3.2 (Disjoint Cycles, Commuting, Product Disjoint Cycles, Product Transposition)

Answer 3

Let n be a positive integer and let S_{n} be the symmetric group of degree n

(a) Disjoint cycles commute
if σ ∩ τ = ∅ then στ = τσ

(b) Every element/permutation in S_{n} can be written as a cycle or a product of disjoint cycles
(c) Every element/permutation in S_{n} can be written as a product of Transpositions

Question 4

Proposition 3.4 (Order, LCM)

Answer 4

Let n be a positive integer, let σ ∈ S_{n}. Then the order of σ is the least common multiple of the lengths of the cycles in a cycle decomposition of σ

Question 5

Even and Odd Permutations

Answer 5

An element of S_{n} that is a finite permutation is either even or odd.

It is an even permutation if it can be written as a product of an even number of transportations.

It is an odd permutation if it can be written as a product of an odd number of transpositions.

Question 6

Lemma 3.6 (product of σ and τ being even or odd)

Answer 6

Let σ, τ ∈ S_{n}
then στ is even if σ, τ are both even or both odd.
στ is odd if one of σ and τ are even and the other is odd

Question 7

The Alternating Group A_{n}

Answer 7

Let n be a positive integer. A_{n} is the alternating group of degree n which is the set of all even permutations in S_{n}