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Subgroup structure of symmetric group:S4

1,969 bytes added, 02:32, 18 December 2010
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| [[subgroup generated by double transposition in S4]] || <math>\{ (), (1,2)(3,4) \}</math> || [[cyclic group:Z2]] || 1 || 3 || -- || 2
|-
| [[A3 in S4]] || <math>\{ (), (1,2,3), (1,3,2) \}</math> || [[cyclic group:Z3]] || 1 || 4 || -- || --
|-
| [[Z4 in S4]] || <math>\langle (1,2,3,4) \rangle</math> || [[cyclic group:Z4]] || 1 || 3 || -- || --
|-
|[[Characteristic subgroup]] || 4 || 4 || 4
|}
 
==Subgroup structure viewed as symmetric group==
 
===Classification based on partition given by orbit sizes===
 
For any subgroup of <math>S_4</math>, the natural action on <math>\{ 1,2,3,4 \}</math> induces a partition of the set <math>\{ 1,2,3 \}</math> into orbits, which in turn induces an unordered integer partition of the number 4. Below, we classify this information for the subgroups.
 
{| class="sortable" border="1"
! Conjugacy class of subgroups !! Size of conjugacy class !! Induced partition of 4 !! Direct product of transitive subgroups on each orbit? !! Illustration with representative
|-
| trivial subgroup || 1 || 1 + 1 + 1 + 1 || Yes || The subgroup fixes each point, so the orbits are singleton subsets.
|-
| [[S2 in S4]] || 6 || 2 + 1 + 1 || Yes || <math>\{ (), (1,2) \}</math> has orbits <math>\{ 1,2 \}, \{ 3 \}, \{ 4 \}</math>
|-
| [[subgroup generated by double transposition in S4]] || 3 || 2 + 2 || No || <math>\{ (), (1,2)(3,4) \}</math> has orbits <math>\{ 1,2 \}, \{ 3, 4 \}</math>
|-
| [[A3 in S4]] || 4 || 3 + 1 || Yes || <math>\{ (), (1,2,3), (1,3,2) \}</math> has orbits <math>\{ 1,2,3 \}, \{ 4 \}</math>
|-
| [[Z4 in S4]] || 3 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.
|-
| [[normal Klein four-subgroup of S4]] || 1 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.
|-
| [[non-normal Klein four-subgroups of S4]] || 3 || 2 + 2 || Yes || <math>\langle (1,2), (3,4) \rangle</math> has orbits <math>\{ 1,2 \}, \{ 3,4 \}</math>
|-
| [[S3 in S4]] || 4 || 3 + 1 || Yes || <math>\langle (1,2,3), (1,2) \rangle</math> has orbits <math>\{ 1,2,3 \}, \{ 4 \}</math>
|-
| [[D8 in S4]] || 3 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.
|-
| [[A4 in S4]] || 1 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.
|-
| whole group || 1 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.
|}
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