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.

Sources: 1 | 2 | 3 | 4

Thursday’s post will be about subgroups! M x

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Set theory is just so wonderful! So many applications

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I’m really enjoying my lectures on it!

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