Editing Group (mathematics) (section)

=== Definition ===
{{quote box
|align =right
|width =33%
|quote =The axioms for a group are short and natural ... Yet somehow hidden behind these axioms is the [[Monster group|monster simple group]], a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.
|author =[[Richard Borcherds]]
|source =''Mathematicians: An Outer View of the Inner World''{{sfn|Cook|2009|p=24}}
}}

A group is a non-empty [[set (mathematics)|set]] <math>G</math> together with a [[binary operation]] on {{tmath|1= G }}, here denoted "{{tmath|1= \cdot }}", that combines any two [[element (mathematics)|elements]] <math>a</math> and <math>b</math> of <math>G</math> to form an element of {{tmath|1= G }}, denoted {{tmath|1= a\cdot b }}, such that the following three requirements, known as '''group axioms''', are satisfied:{{sfn|Artin|2018|loc=§2.2|p=40}}{{sfn|Lang|2002|loc = p. 3, I.§1 and p. 7, I.§2}}{{sfn|Lang|2005|loc=II.§1|p=16}}{{efn|Some authors include an additional axiom referred to as the ''closure'' under the operation <!--use {{math}}, since <math> in footnotes is unreadable on mobile devices-->"{{math|⋅}}", which means that {{math|''a'' ⋅ ''b''}} is an element of {{math|''G''}} for every {{math|''a''}} and {{math|''b''}} in {{math|''G''}}. This condition is subsumed by requiring "{{math|⋅}}" to be a binary operation on {{math|''G''}}. See {{harvard citations|nb = yes|last = Lang|year = 2002}}.}}

; Associativity : For all {{tmath|1= a }}, {{tmath|1= b }}, {{tmath|1= c }} in {{tmath|1= G }}, one has {{tmath|1= (a\cdot b)\cdot c=a\cdot(b\cdot c) }}.
; Identity element : There exists an element <math>e</math> in <math>G</math> such that, for every <math>a</math> in {{tmath|1= G }}, one has {{tmath|1= e\cdot a=a }} and {{tmath|1= a\cdot e=a }}.
: Such an element is unique ([[#Uniqueness of identity element|see below]]). It is called the ''identity element'' (or sometimes ''neutral element'') of the group.
; Inverse element : For each <math>a</math> in {{tmath|1= G }}, there exists an element <math>b</math> in <math>G</math> such that <math>a\cdot b=e</math> and {{tmath|1= b\cdot a=e }}, where <math>e</math> is the identity element.
: For each {{tmath|1= a }}, the element <math>b</math> is unique ([[#Uniqueness of inverses|see below]]); it is called ''the inverse'' of <math>a</math> and is commonly denoted {{tmath|1= a^{-1} }}.