This week I wanted to discuss a topic that I’ve been learning about in lectures: groups. In this post I will highlight the axioms that a set of numbers must satisfy in order for it to be a group, i.e. the definition of a group.
So, a group (G,∘) (where ∘ is the operation) must satisfy the following conditions:
- Closure: if a and b are two elements in G, then a∘b is also an element in G
∀ a,b ∈ G : a∘b ∈ G
- Associativity: the defined operation is associative
∀ a,b,c ∈ G : a∘(b∘c) = (a∘b)∘c
- Identity: there is an identity element e which is in G
∃ e ∈ G : ∀ a ∈ G: e∘a = a = a∘e
- Inverse: each element a in G must have an inverse b that is also in G
∀ a ∈ G : ∃ b ∈ G : a∘b = e = b∘a
A group is abelian if it is commutative, i.e.∀ a,b ∈ G: a•b = b•a.
EXAMPLE: Prove (Z,+) is a group.
To prove that something is a group, we must go through the group axioms and check that all of them hold.
- For any two integers a, b, the sum of these integers a+b is also an integer. Therefore the set is closed under addition.
- For all integers a, b, c, (a+b) + c = a + (b+c), hence associativity is satisfied.
- If a is an integer, then a + 0 = a = 0 + a, so the identity element is 0, which is also an integer. Therefore, this set contains the identity element 0.
- For every integer a, there is an integer b such that a + b = 0 = b + a, as this occurs when b = -a. Hence, each number in the set has an inverse element which is also in the set of integers.
Concluding, it follows that the set of integers is a group under addition. Note that this group is also abelian as a + b = b + a for all integers a, b.
Thursday’s post will be about subgroups! M x