8) More on permutations

Question 1

How can every permutation in Sn be expressed

Answer 1

Every permutation f ∈ Sn is a product of transpositions (i.e. 2-cycles)

Question 2

What are even/odd permutations

Answer 2

Let f ∈ Sn. We say that f is even if it can be written as a product
of an even number of transpositions, and similarly, f is odd if it’s a product of an odd number of transpositions

Question 3

What is the alternating group of degree n

Answer 3

The subgroup of Sn consisting of even permutations

Question 4

What is the Parity Theorem

Answer 4

Let f ∈ Sn. Then f is either even or odd but not both

Question 5

What is the sign of a permutation

Answer 5

Question 6

What is sign the homomophorism of

Answer 6

Sn → {1, −1} is a surjective homomorphism onto the group {1, −1} with binary operation given by multiplication

Question 7

What is the order of the alternating group A for n≥2

Answer 7

Question 8

What is the index of H∩A in H for any subgroup H of Sn

Answer 8

Let H ≤ Sn. Then the index [H : H ∩ An] is equal to either 1 or 2. i.e.
for any subgroup H ≤ Sn the even permutations of H either make up all of H or half of H

Question 9

How do the centraliser and conjugacy class sizes relate in Sn and An

Answer 9

Question 10

What are the relationships between centralisers and conjugacy classes in Sn and An

Answer 10

Question 11

How does a permutation conjugate a cycle in Sn

Answer 11