Editing Maths (section)

==='''Group'''===
An algebraic structure that is a set with a single binary operation.

An '''infinite group''' has an unlimited number of elements. (ℤ, +)

A '''finite groups''' has a finite number of elements. (ℤₘ, +) - The integers mod m.

The order of an element is the number of elements in the subgroup it generates. |2| = 3 

* '''(𝔾, ∗)''' - Group '''𝔾''' with operation '''∗'''.

* Axioms
** The operation must be '''closed'''. - '''a,b ∈ 𝔾 → a∗b ∈ 𝔾'''
** The operation must be '''associative'''. - '''(a∗b)∗c = a∗(b∗c)'''
** Operation must have an '''identity''' element that has no effect on any other element under operation. '''∃e(a∗e = e∗a = a)'''.
** Every element must have an '''inverse'''. An element when combined with the original will return the identity. '''∀a∃a⁻¹(a∗a⁻¹=a⁻¹∗a=e)'''

* '''(ℤ, +)''' - The integers under addition.
** Closure: '''a,b ∈ ℤ → a+b ∈ ℤ'''
** Associativity: '''(a+b)+c = a+(b+c)'''
** Identity: '''a+0 = 0+a = a'''
** Inverse: '''a⁻¹ = -a'''

* '''(ℤ, ×)''' - The integers under multiplication it '''not a group'''
** Closure: '''a,b ∈ ℤ → a×b ∈ ℤ'''
** Associativity: '''(a×b)×c = a×(b×c)'''
** Identity: '''a×1 = 1×a = a'''
** '''NO''' Inverse: '''2x=1 has no solution in ℤ'''

* If the operation is also '''commutative''' then it is an '''Abelian''' group.
* [https://www.youtube.com/watch?v=WwndchnEDS4 Bill Shillito].