S2, C7: Cosets and Lagrange's Theorem

Question 1

Let G be a group and H be a subgroup of G. Let g be an

element of G. What is a left coset of H in G?

Answer 1

gH = {gh : h ∈ H}

Question 2

Let G be a group, let H be a subgroup of G and let a, b ∈ G.

We say that a is equivalent to b modulo H, and write a ≡ b mod H, iff

Answer 2

b^(−1)a ∈ H

Question 3

Let G be a group with a subgroup H and let g∈G. Then, what link the size of gH and the size of H?

Answer 3

|gH| = |H|

Question 4

What is Lagrange’s Theorem?

Answer 4

Let G be a finite group and let H be a subgroup of G. The order of G is a multiple of the order of H.
|G| = m |H| where m is the number of distinct left cosets of H in G.

Question 5

Let G be a finite group and let a∈G. Then,

1. the order of G is?
2. a^(|G|) = ?

Answer 5

1. a multiple of the order of a.

2. e