Groups & Fields

What is a binary operation on a set V to a set C?

A function B : V × V → C.

What is an internal binary operation on a set V?

A binary operation for which C = V , or equivalently, x₁ ∗ x₂ ∈ V ∀x₁, x₂ ∈ V.

If a binary operation is internal, then the set V is said to be closed under the binary operation.

What does it mean if a binary operation on a set V is commutative?

Iff for all x₁, x₂ ∈ V , x₁ ∗ x₂ = x₂ ∗ x₁

What does it mean if a binary operation on a set V is associative?

Iff for all x₁, x₂, x₃∈ V , (x₁ ∗ x₂) ∗ x₃= x₁ ∗ (x₂ ∗ x₃)

What does it mean if an element e ∈ V is called an identity for a binary operation on a set V?

Iff for all x₁ ∈ V, e ∗ x₁ = x₁ ∗ e = x₁

What is the proof for the theorem ‘If a binary operation on a set V has an identity, then the identity is unique.’

Assume there are two elements e₁ and e₂ in V which are both identities. Then, e₁ ∗ e₂ = e₂ since e₁ is an identity. However, e₁ ∗ e₂ = e₁ since e₂ is an identity. Hence, e₁ = e₂, as required.

Consider a binary operation on a set V which has identity e ∈ V. What does it mean if an element a ∈ V has an inverse?

If there exists an element b ∈ V such that a ∗ b = b ∗ a = e.

When is a set V, endowed with an internal binary operation ∗, called a group with respect to the binary operation ∗?

Iff the following hold:

1. ∗ is associative;

2. ∗ has an identity V

3. Every element of V have an inverse element.

What does it mean if a group with binary operation ∗ is called abelian?

Iff ∗ is commutative.

What does it mean if a set F with binary operations + and × is called a field (F, +, ×)?

Iff:

1. (F, +) is an abelian group, with identity denoted by 0;

2. (F \ {0}, ×) is an abelian group, with identity denoted by 1;

3. for all x ∈ F, 0 × x = x × 0 = 0; and

4. distributivity of × with respect to +: for all x₁, x₂, x₃ ∈ F, we have

x₁ × (x₂ + x₃) = (x₁ × x₂) + (x₁ × x₃).