# Changes

## Subgroup structure of symmetric group:S4

, 02:32, 18 December 2010
no edit summary
| [[subgroup generated by double transposition in S4]] || $\{ (), (1,2)(3,4) \}$ || [[cyclic group:Z2]] || 1 || 3 || -- || 2
|-
| [[A3 in S4]] || $\{ (), (1,2,3), (1,3,2) \}$ || [[cyclic group:Z3]] || 1 || 4 || -- || --
|-
| [[Z4 in S4]] || $\langle (1,2,3,4) \rangle$ || [[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 $S_4$, the natural action on $\{ 1,2,3,4 \}$ induces a partition of the set $\{ 1,2,3 \}$ 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 || $\{ (), (1,2) \}$ has orbits $\{ 1,2 \}, \{ 3 \}, \{ 4 \}$
|-
| [[subgroup generated by double transposition in S4]] || 3 || 2 + 2 || No || $\{ (), (1,2)(3,4) \}$ has orbits $\{ 1,2 \}, \{ 3, 4 \}$
|-
| [[A3 in S4]] || 4 || 3 + 1 || Yes || $\{ (), (1,2,3), (1,3,2) \}$ has orbits $\{ 1,2,3 \}, \{ 4 \}$
|-
| [[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 || $\langle (1,2), (3,4) \rangle$ has orbits $\{ 1,2 \}, \{ 3,4 \}$
|-
| [[S3 in S4]] || 4 || 3 + 1 || Yes || $\langle (1,2,3), (1,2) \rangle$ has orbits $\{ 1,2,3 \}, \{ 4 \}$
|-
| [[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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