Editing Group (mathematics) (section)

=== 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.