Editing Cayley's theorem (section)

==Examples of the regular group representation==
<math> \mathbb Z_2 = \{0,1\} </math> with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12) (see [[Permutation#Cycle_notation|cycle notation]]). E.g. 0 +1 = 1 and 1+1 = 0, so <math display=inline>1\mapsto0</math> and <math display=inline>0\mapsto1,</math> as they would under a permutation.

<math> \mathbb Z_3 = \{0,1,2\} </math> with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132). E.g. 1&nbsp;+&nbsp;1&nbsp;=&nbsp;2 corresponds to (123)(123)&nbsp;=&nbsp;(132).

<math> \mathbb Z_4 = \{0,1,2,3\} </math> with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).

The elements of [[Klein four-group]] {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).

S<sub>3</sub> ([[dihedral group of order 6]]) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements, and the latter is how it is realized by its regular representation.

<!-- Looks ugly if it's left-aligned/non-square cells, so a bit of customization is good here -->
{| class="wikitable" style="text-align: center;"
! style="width: 1.5em; height: 1.5em;" | *
! style="width: 1.5em;" | ''e''
! style="width: 1.5em;" | ''a''
! style="width: 1.5em;" | ''b''
! style="width: 1.5em;" | ''c''
! style="width: 1.5em;" | ''d''
! style="width: 1.5em;" | ''f''
! permutation
|-
! style="height: 1.5em;" | ''e''
| ''e'' || ''a'' || ''b'' || ''c'' || ''d'' || ''f'' || ''e''
|-
! style="height: 1.5em;" | ''a''
| ''a'' || ''e'' || ''d'' || ''f'' || ''b'' || ''c'' || (12)(35)(46)
|-
! style="height: 1.5em;" | ''b''
| ''b'' || ''f'' || ''e'' || ''d'' || ''c'' || ''a'' || (13)(26)(45)
|-
! style="height: 1.5em;" | ''c''
| ''c'' || ''d'' || ''f'' || ''e'' || ''a'' || ''b'' || (14)(25)(36)
|-
! style="height: 1.5em;" | ''d''
| ''d'' || ''c'' || ''a'' || ''b'' || ''f'' || ''e'' || (156)(243)
|-
! style="height: 1.5em;" | ''f''
| ''f'' || ''b'' || ''c'' || ''a'' || ''e'' || ''d'' || (165)(234)
|}