Editing Subgroup (section)

==Example: Subgroups of Z<sub>8</sub>==<!-- This section is linked from [[List of small groups]] -->
Let {{mvar|G}} be the [[cyclic group]] {{math|Z<sub>8</sub>}} whose elements are
:<math>G = \left\{0, 4, 2, 6, 1, 5, 3, 7\right\}</math>
and whose group operation is [[modular arithmetic|addition modulo 8]]. Its [[Cayley table]] is

{| class="wikitable" style="color:blue;"
|-
! style="background-color:#FFFFAA; color:black;" | +
! style="background-color:#FFFFAA; color:orange;" | 0
! style="background-color:#FFFFAA; color:orange;" | 4
! style="background-color:#FFFFAA; color:red;" | 2
! style="background-color:#FFFFAA; color:red;" | 6
! style="background-color:#FFFFAA;" | 1
! style="background-color:#FFFFAA;" | 5
! style="background-color:#FFFFAA;" | 3
! style="background-color:#FFFFAA;" | 7
|-
! style="background:#FFFFAA; color:orange;" | 0
| style="color:orange;" | 0 || style="color:orange;" | 4
| style="color:red;"    | 2 || style="color:red;"    | 6
| 1 || 5 || 3 || 7
|-
! style="background:#FFFFAA; color:orange;" | 4
| style="color:orange;" | 4 || style="color:orange;" | 0
| style="color:red;"    | 6 || style="color:red;"    | 2
| 5 || 1 || 7 || 3
|-
! style="background:#FFFFAA; color:red;" | 2
| style="color:red;" | 2 ||  style="color:red;" | 6 || style="color:red;" | 4 || style="color:red;" | 0
| 3 || 7 || 5 || 1
|-
! style="background:#FFFFAA; color:red;" | 6
| style="color:red;" | 6 || style="color:red;" | 2 || style="color:red;" | 0 || style="color:red;" | 4
| 7 || 3 || 1 || 5
|-
! style="background-color:#FFFFAA;" | 1
| 1 || 5 || 3 || 7 || 2 || 6 || 4 || 0
|-
! style="background-color:#FFFFAA;" | 5
| 5 || 1 || 7 || 3 || 6 || 2 || 0 || 4
|-
! style="background-color:#FFFFAA;" | 3
| 3 || 7 || 5 || 1 || 4 || 0 || 6 || 2
|-
! style="background-color:#FFFFAA;" | 7
| 7 || 3 || 1 || 5 || 0 || 4 || 2 || 6
|}

This group has two nontrivial subgroups: {{math|{{colorbull|orange}} ''J'' {{=}} {0, 4} }} and {{math|{{colorbull|red}} ''H'' {{=}} {0, 4, 2, 6} }}, where {{mvar|J}} is also a subgroup of {{mvar|H}}.  The Cayley table for {{mvar|H}} is the top-left quadrant of the Cayley table for {{mvar|G}}; The Cayley table for {{mvar|J}} is the top-left quadrant of the Cayley table for {{mvar|H}}.  The group {{mvar|G}} is [[cyclic group|cyclic]], and so are its subgroups.  In general, subgroups of cyclic groups are also cyclic.{{sfn|Gallian|2013|p=81}}