Editing Group (mathematics) (section)

== Definition and illustration ==

=== First example: the integers ===
One of the more familiar groups is the set of [[integer]]s 
<math display=block>\Z = \{\ldots,-4,-3,-2,-1,0,1,2,3,4,\ldots\}</math>
together with [[addition]].{{sfn|Lang|2005|loc=App. 2|p=360}} For any two integers <math>a</math> and {{tmath|1= b }}, the [[Summation|sum]] <math>a+b</math> is also an integer; this ''[[Closure (mathematics)|closure]]'' property says that <math>+</math> is a [[binary operation]] on {{tmath|1= \Z }}. The following properties of integer addition serve as a model for the group axioms in the definition below.
* For all integers {{tmath|1= a }}, <math>b</math> and&nbsp;{{tmath|1= c }}, one has {{tmath|1= (a+b)+c=a+(b+c) }}. Expressed in words, adding <math>a</math> to <math>b</math> first, and then adding the result to <math>c</math> gives the same final result as adding <math>a</math> to the sum of <math>b</math> and&nbsp;{{tmath|1= c }}. This property is known as ''[[associativity]]''.
* If <math>a</math> is any integer, then <math>0+a=a</math> and {{tmath|1= a+0=a }}. [[Zero]] is called the ''[[identity element]]'' of addition because adding it to any integer returns the same integer.
* For every integer {{tmath|1= a }}, there is an integer <math>b</math> such that <math>a+b=0</math> and {{tmath|1= b+a=0 }}. The integer <math>b</math> is called the ''[[inverse element]]'' of the integer <math>a</math> and is denoted&nbsp;{{tmath|1= -a }}.

The integers, together with the operation {{tmath|1= + }}, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.

=== 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} }}.

=== Notation and terminology ===
Formally, a group is an [[ordered pair]] of a set and a binary operation on this set that satisfies the [[group axioms]]. The set is called the ''underlying set'' of the group, and the operation is called the ''group operation'' or the ''group law''.

A group and its underlying set are thus two different [[mathematical object]]s. To avoid cumbersome notation, it is common to [[abuse of notation|abuse notation]] by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.

For example, consider the set of [[real number]]s {{tmath|1= \R }}, which has the operations of addition <math>a+b</math> and [[multiplication]] {{tmath|1= ab }}. Formally, <math>\R</math> is a set, <math>(\R,+)</math> is a group, and <math>(\R,+,\cdot)</math> is a [[field (mathematics)|field]]. But it is common to write <math>\R</math> to denote any of these three objects.

The ''additive group'' of the field <math>\R</math> is the group whose underlying set is <math>\R</math> and whose operation is addition. The ''multiplicative group'' of the field <math>\R</math> is the group <math>\R^{\times}</math> whose underlying set is the set of nonzero real numbers <math>\R \smallsetminus \{0\}</math> and whose operation is multiplication.

More generally, one speaks of an ''additive group'' whenever the group operation is notated as addition; in this case, the identity is typically denoted {{tmath|1= 0 }}, and the inverse of an element <math>x</math> is denoted {{tmath|1= -x }}. Similarly, one speaks of a ''multiplicative group'' whenever the group operation is notated as multiplication; in this case, the identity is typically denoted {{tmath|1= 1 }}, and the inverse of an element <math>x</math> is denoted {{tmath|1= x^{-1} }}. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, <math>ab</math> instead of {{tmath|1= a\cdot b }}.

The definition of a group does not require that <math>a\cdot b=b\cdot a</math> for all elements <math>a</math> and <math>b</math> in {{tmath|1= G }}. If this additional condition holds, then the operation is said to be [[commutative]], and the group is called an [[abelian group]]. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.

Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are [[function (mathematics)|functions]], the operation is often [[function composition]] {{tmath|1= f\circ g }}; then the identity may be denoted id. In the more specific cases of [[geometric transformation]] groups, [[symmetry (mathematics)|symmetry]] groups, [[permutation group]]s, and [[automorphism group]]s, the symbol <math>\circ</math> is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.

=== Second example: a symmetry group ===
Two figures in the [[plane (geometry)|plane]] are [[congruence (geometry)|congruent]] if one can be changed into the other using a combination of [[rotation (mathematics)|rotation]]s, [[reflection (mathematics)|reflection]]s, and [[translation (geometry)|translation]]s. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called [[symmetry|symmetries]]. A [[square]] has eight symmetries. These are:
{| class="wikitable" style="text-align:center;"
|+ The elements of the symmetry group of the square, {{tmath|1= \mathrm{D}_4 }}. Vertices are identified by color or number.
|-
| [[Image:group D8 id.svg|140px|class=skin-invert-image|alt=A square with its four corners marked by 1 to 4]] {{br}} <math>\mathrm{id}</math> (keeping it as it is) || [[Image:group D8 90.svg|140px|class=skin-invert-image|alt=The square is rotated by 90° clockwise; the corners are enumerated accordingly.]] {{br}} <math>r_1</math> (rotation by 90° clockwise) || [[Image:group D8 180.svg|140px|class=skin-invert-image|alt=The square is rotated by 180° clockwise; the corners are enumerated accordingly.]] {{br}} <math>r_2</math> (rotation by 180°) || [[Image:group D8 270.svg|140px|class=skin-invert-image|alt=The square is rotated by 270° clockwise; the corners are enumerated accordingly.]] {{br}} <math>r_3</math> (rotation by 270° clockwise)
|-
| [[Image:group D8 fv.svg|140px|class=skin-invert-image|alt=The square is reflected vertically; the corners are enumerated accordingly.]] {{br}} <math>f_{\mathrm{v}}</math> (vertical reflection) || 
[[Image:group D8 fh.svg|140px|class=skin-invert-image|alt=The square is reflected horizontally; the corners are enumerated accordingly.]] {{br}} <math>f_{\mathrm{h}}</math> (horizontal reflection) 
|| 
[[Image:group D8 f13.svg|140px|class=skin-invert-image|alt=The square is reflected along the SW–NE diagonal; the corners are enumerated accordingly.]] {{br}} <math>f_{\mathrm{d}}</math> (diagonal reflection) 
|| 
[[Image:group D8 f24.svg|140px|class=skin-invert-image|alt=The square is reflected along the SE–NW diagonal; the corners are enumerated accordingly.]] {{br}} <math>f_{\mathrm{c}}</math> (counter-diagonal reflection)
|}
* the [[identity operation]] leaving everything unchanged, denoted id;
* rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by {{tmath|1= r_1 }}, <math>r_2</math> and {{tmath|1= r_3 }}, respectively;
* reflections about the horizontal and vertical middle line ({{tmath|1= f_{\mathrm{v} } }} and {{tmath|1= f_{\mathrm{h} } }}), or through the two [[diagonal]]s ({{tmath|1= f_{\mathrm{d} } }} and {{tmath|1= f_{\mathrm{c} } }}).
{{clear}}

These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, <math>r_1</math> sends a point to its rotation 90° clockwise around the square's center, and <math>f_{\mathrm{h}}</math> sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the [[dihedral group]] of degree four, denoted {{tmath|1= \mathrm{D}_4 }}. The underlying set of the group is the above set of symmetries, and the group operation is function composition.{{sfn|Herstein|1975|loc=§2.6|p=54}} Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first <math>a</math> and then <math>b</math> is written symbolically ''from right to left'' as <math>b\circ a</math> ("apply the symmetry <math>b</math> after performing the symmetry {{tmath|1= a }}"). This is the usual notation for composition of functions.

A [[Cayley table]] lists the results of all such compositions possible. For example, rotating by 270° clockwise ({{tmath|1= r_3 }}) and then reflecting horizontally ({{tmath|1= f_{\mathrm{h} } }}) is the same as performing a reflection along the diagonal ({{tmath|1= f_{\mathrm{d} } }}). Using the above symbols, highlighted in blue in the Cayley table:
<math display=block>f_\mathrm h \circ r_3= f_\mathrm d.</math>

{| class="wikitable" style="float:right; text-align:center; margin:.5em 0 .5em 1em; width:40ex; height:40ex;"
|+ [[Cayley table]] of <math>\mathrm{D}_4</math>
|-
!  style="width:12%; background:#fdd; border-top:solid black 2px; border-left:solid black 2px;"| <math>\circ</math>
!  style="background:#fdd; border-top:solid black 2px; width:11%;"| <math>\mathrm{id}</math>
!  style="background:#fdd; border-top:solid black 2px; width:11%;"| <math>r_1</math>
!  style="background:#fdd; border-top:solid black 2px; width:11%;"| <math>r_2</math>
!  style="background:#fdd; border-right:solid black 2px; border-top:solid black 2px; width:11%;"| <math>r_3</math>
! style="width:11%;"| <math>f_{\mathrm{v}}</math> !! style="width:11%;"| <math>f_{\mathrm{h}}</math> !! style="width:11%;"| <math>f_{\mathrm{d}}</math> !! style="width:11%;"| <math>f_{\mathrm{c}}</math>
|-
!style="background:#FDD;  border-left:solid black 2px;" | <math>\mathrm{id}</math>
|style="background:#FDD;"| <math>\mathrm{id}</math>
|style="background:#FDD;"| <math>r_1</math>
|style="background:#FDD;" | <math>r_2</math>
|style="background:#FDD; border-right:solid black 2px;"| <math>r_3</math> || <math>f_{\mathrm{v}}</math> || <math>f_{\mathrm{h}}</math> || <math>f_{\mathrm{d}}</math>
|style="background:#FFFC93; border-right:solid black 2px; border-left:solid black 2px; border-top:solid black 2px;"| <math>f_{\mathrm{c}}</math>
|-
!style="background:#FDD;  border-left:solid black 2px;" | <math>r_1</math>
|style="background:#FDD;"| <math>r_1</math>
|style="background:#FDD;"| <math>r_2</math>
|style="background:#FDD;"| <math>r_3</math>
|style="background:#FDD; border-right:solid black 2px;"| <math>\mathrm{id}</math> || <math>f_{\mathrm{c}}</math> || <math>f_{\mathrm{d}}</math> || <math>f_{\mathrm{v}}</math>
|style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px;"| <math>f_{\mathrm{h}}</math>
|- style="height:10%"
!style="background:#FDD;  border-left:solid black 2px;" | <math>r_2</math>
|style="background:#FDD;"| <math>r_2</math>
|style="background:#FDD;"| <math>r_3</math>
|style="background:#FDD;"| <math>\mathrm{id}</math>
|style="background:#FDD; border-right:solid black 2px;"| <math>r_1</math> || <math>f_{\mathrm{h}}</math> || <math>f_{\mathrm{v}}</math> || <math>f_{\mathrm{c}}</math>
|style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px;"| <math>f_{\mathrm{d}}</math>
|- style="height:10%"
!style="background:#FDD; border-bottom:solid black 2px; border-left:solid black 2px;" | <math>r_3</math>
|style="background:#FDD; border-bottom:solid black 2px;"| <math>r_3</math>
|style="background:#FDD; border-bottom:solid black 2px;"| <math>\mathrm{id}</math>
|style="background:#FDD; border-bottom:solid black 2px;"| <math>r_1</math>
|style="background:#FDD; border-right:solid black 2px; border-bottom:solid black 2px;"| <math>r_2</math> ||  <math>f_{\mathrm{d}}</math> ||  <math>f_{\mathrm{c}}</math> ||  <math>f_{\mathrm{h}}</math>
|style="background:#FFFC93; border-right:solid black 2px; border-left:solid black 2px; border-bottom:solid black 2px;"|  <math>f_{\mathrm{v}}</math>
|- style="height:10%"
! <math>f_{\mathrm{v}}</math>
| <math>f_{\mathrm{v}}</math> || <math>f_{\mathrm{d}}</math> || <math>f_{\mathrm{h}}</math> || <math>f_{\mathrm{c}}</math> || <math>\mathrm{id}</math> || <math>r_2</math> || <math>r_1</math> || <math>r_3</math>
|- style="height:10%"
! <math>f_{\mathrm{h}}</math>
| <math>f_{\mathrm{h}}</math> || <math>f_{\mathrm{c}}</math> || <math>f_{\mathrm{v}}</math>
|style="background:#BACDFF; border: solid black 2px;"| <math>f_{\mathrm{d}}</math>
|"style=background:#FFFC93;"| <math>r_2</math> || <math>\mathrm{id}</math> || <math>r_3</math> || <math>r_1</math>
|- style="height:10%"
! <math>f_{\mathrm{d}}</math>
| <math>f_{\mathrm{d}}</math> || <math>f_{\mathrm{h}}</math> || <math>f_{\mathrm{c}}</math> || <math>f_{\mathrm{v}}</math> || <math>r_3</math> || <math>r_1</math> || <math>\mathrm{id}</math> || <math>r_2</math>
|- style="height:10%"
! <math>f_{\mathrm{c}}</math>
|style="background:#9DFF93; border-left: solid black 2px; border-bottom: solid black 2px; border-top: solid black 2px;" | <math>f_{\mathrm{c}}</math>
|style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px;" | <math>f_{\mathrm{v}}</math>
|style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px;" | <math>f_{\mathrm{d}}</math>
|style="background:#9DFF93; border-bottom:solid black 2px; border-top:solid black 2px; border-right:solid black 2px;" | <math>f_{\mathrm{h}}</math> || <math>r_1</math> || <math>r_3</math> || <math>r_2</math> || <math>\mathrm{id}</math>
|-
| colspan="9" style="text-align:left"| The elements {{tmath|1= \mathrm{id} }}, {{tmath|1= r_1 }}, {{tmath|1= r_2 }}, and {{tmath|1= r_3 }} form a [[subgroup]] whose Cayley table is highlighted in {{color box|#FDD}} red (upper left region). A left and right [[coset]] of this subgroup are highlighted in {{color box|#9DFF93}} green (in the last row) and {{color box|#FFFC93}} yellow (last column), respectively. The result of the composition {{tmath|1= f_{\mathrm{h} }\circ r_3 }}, the symmetry {{tmath|1= f_{\mathrm{d} } }}, is highlighted in {{color box|#BACDFF}} blue (below table center).
|}
Given this set of symmetries and the described operation, the group axioms can be understood as follows.

''Binary operation'': Composition is a binary operation. That is, <math>a\circ b</math> is a symmetry for any two symmetries <math>a</math> and {{tmath|1= b }}. For example,
<math display=block>r_3\circ f_\mathrm h = f_\mathrm c,</math>
that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal ({{tmath|1= f_{\mathrm{c} } }}). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the Cayley table.

''Associativity'': The associativity axiom deals with composing more than two symmetries: Starting with three elements {{tmath|1= a }}, {{tmath|1= b }} and {{tmath|1= c }} of {{tmath|1= \mathrm{D}_4 }}, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose <math>a</math> and <math>b</math> into a single symmetry, then to compose that symmetry with {{tmath|1= c }}. The other way is to first compose <math>b</math> and {{tmath|1= c }}, then to compose the resulting symmetry with {{tmath|1= a }}. These two ways must give always the same result, that is,
<math display=block>(a\circ b)\circ c = a\circ (b\circ c),</math>
For example, <math>(f_{\mathrm{d}}\circ f_{\mathrm{v}})\circ r_2=f_{\mathrm{d}}\circ (f_{\mathrm{v}}\circ r_2)</math> can be checked using the Cayley table:
<math display=block>\begin{align}
(f_\mathrm d\circ f_\mathrm v)\circ r_2 &=r_3\circ r_2=r_1\\
f_\mathrm d\circ (f_\mathrm v\circ r_2) &=f_\mathrm d\circ f_\mathrm h =r_1.
\end{align}</math>

''Identity element'': The identity element is {{tmath|1= \mathrm{id} }}, as it does not change any symmetry <math>a</math> when composed with it either on the left or on the right.

''Inverse element'': Each symmetry has an inverse: {{tmath|1= \mathrm{id} }}, the reflections {{tmath|1= f_{\mathrm{h} } }}, {{tmath|1= f_{\mathrm{v} } }}, {{tmath|1= f_{\mathrm{d} } }}, {{tmath|1= f_{\mathrm{c} } }} and the 180° rotation <math>r_2</math> are their own inverse, because performing them twice brings the square back to its original orientation. The rotations <math>r_3</math> and <math>r_1</math> are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.

In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in {{tmath|1= \mathrm{D}_4 }}, as, for example, <math>f_{\mathrm{h}}\circ r_1=f_{\mathrm{c}}</math> but {{tmath|1= r_1\circ f_{\mathrm{h} }=f_{\mathrm{d} } }}. In other words, <math>\mathrm{D}_4</math> is not abelian.