M1: Groups & Group Actions

Question 1

Proposition

Subgroup Test

Answer 1

Let G be a group and H ⊆ G be non-empty
Then H is a subgroup of G denoted by H ≤ G if and only if
∀ x, y ∈ H: x⁻¹y ∈ H

Question 2

Proof

(Subgroup Test)
Let G be a group and H ⊆ G be non-empty
Then H is a subgroup of G denoted by H ≤ G if and only if
∀ x, y ∈ H: x⁻¹y ∈ H

1 point

Answer 2

• (⇐) Check off subgroup axioms

Question 3

Theorem

Lagrange’s Theorem

Answer 3

Let G be a finite group with a subgroup H. Then |H| | |G|
In fact |G| = |H||G/H|

Question 4

Proof

(Lagrange’s Theorem)
Let G be a finite group with a subgroup H. Then |H| | |G|
In fact |G| = |H||G/H|

2 points

Answer 4

• |gH| = |H| by defining bijection
• G/H partitions G

Question 5

Theorem

Fermat’s Little Theorem

Answer 5

Let p be a prime number and let a ∈ ℤ s.t. p ∤ a
Then aᵖ⁻¹ ≡ 1 mod p

Question 6

Proof

(Fermat’s Little Theorem)
Let p be a prime number and let a ∈ ℤ s.t. p ∤ a
Then aᵖ⁻¹ ≡ 1 mod p

2 points

Answer 6

• Consider G = ℤ*ₚ
• a̅ᵖ⁻¹ = 1̅

Question 7

Theorem

Euler’s Theorem

Answer 7

Let n ∈ ℕ
Let a ∈ ℤ such that a is coprime to n
Then a^(φ(n)) ≡ 1 mod n