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Subgroup structure of symmetric group:S4

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# The trivial subgroup. Isomorphic to [[subgroup::trivial group]].(1)
# The two-element subgroup generated by a transposition, such as <math>(1,2)</math>. Isomorphic to [[subgroup::cyclic group of order two]].(6)
# The two-element subgroup generated by a double transposition, such as <math>(1,2)(3,4)</math>. Isomorphic to [[subgroup::cyclic group:Z2|cyclic group of order two]]. (3)
# The four-element subgroup generated by two disjoint transpositions, such as <math>\langle (1,2) \ , \ (3,4) \rangle</math>. Isomorphic to [[subgroup::Klein-four group]]. (3)
# The unique four-element subgroup comprising the identity and the three double transpositions. Isomorphic to [[subgroup::Klein-four group]]. (1)
# The four-element subgroup spanned by a 4-cycle. Isomorphic to [[subgroup::cyclic group:Z4|cyclic group of order four]].(3)# The eight-element subgroup spanned by a 4-cycle and a transposition that conjugates this cycle to its inverse. Isomorphic to [[subgroup::dihedral group:D8|dihedral group of order eight]]. This is also a 2-Sylow subgroup. (3)# The three-element subgroup spanned by a three-cycle. Isomorphic to [[subgroup::cyclic group:Z3|cyclic group of order three]].(4)
# The six-element subgroup comprising all permutations that fix one element. Isomorphic to [[subgroup::symmetric group:S3|symmetric group on three elements]]. (4)
# The alternating group: the subgroup of all even permutations. Isomorphic to [[subgroup::alternating group:A4]].(1)
<math>\{ (), (1,2)(3,4) \}, \{ (1,3)(2,4) \}, \{ (1,4)(2,3) \}</math>.
These three subgroups are in bijection with the three subgroups of type (4) via the centralizer operation, and also with the three 2-Sylow subgroups of order eight (type (7)) via the normalizer operation.
===Subgroup properties satisfied by these subgroups===
* [[Normal closure]]: The normal closure is the Klein-four group, type (5).
* [[Normalizer]]: The normalizer is a dihedral group. Each of these subgroups has a different dihedral group as its normalizer.
* [[Centralizer]]: The centralizer is the same as the normal closurenormalizer.
==The non-normal Klein-four groups (type (4))==
There are three such groups. These three groups are in bijection with the three subgroups of type (3) discussed above via the centralizer operation, and also with the three 2-Sylow subgroups, via the normalizer operation.
<math>\{ (), (1,2), (3,4), (1,2)(3,4) \}, \{ (), (1,3), (2,4), (1,3)(2,4) \}, \{ (), (1,4), (2,3), (1,4)(2,3) \}</math>
* [[Permutable subgroup]]
* [[Conjugate-permutable subgroup]]
 
===Effect of operators===
 
* [[Normal closure]]: The normal closure of any of these is the whole group.
* [[Normalizer]]: The normalizer of any of these is a dihedral group of order eight (type (7)). This establishes a bijection between the subgroups of type (4) and the subgroups of type (7).
* [[Centralizer]]: The centralizer equals the subgroup itself.
 
==Cyclic subgroup of order four (type (6))==
 
There are three such subgroups, given as follows:
 
<math>\{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) \}, \{ (), (1,2,4,3), (1,4)(3,2), (1,3,4,2) \}, \{ (), (1,3,2,4), (1,2)(3,4), (1,4,2,3)\}</math>.
 
===Subgroup properties satisfied by these subgroups===
 
* [[Contranormal subgroup]]
* [[Core-free subgroup]]
* [[Self-centralizing subgroup]]
* [[Pronormal subgroup]]
* [[Isomorph-conjugate subgroup]]
* [[Intermediately isomorph-conjugate subgroup]]
 
===Subgroup properties not satisfied by these subgroups===
 
* [[Self-normalizing subgroup]]
* [[Subnormal subgroup]]
 
===Effect of operators===
 
* [[Normalizer]]: The normalizer of any cyclic subgroup is a dihedral group of order eight. In fact, this establishes a bijection between the three cyclic subgroups of order four (type (6)) and the three dihedral subgroups of order eight (type (7)).
* [[Centralizer]]: Each subgroup is its own centralizer.
* [[Normal closure]]: The normal closure of each such subgroup is the whole group.
 
===Other facts about this subgroup===
 
This subgroup of the symmetric group on four elements can be obtained by starting with an abstract [[cyclic group:Z4|cyclic group of order four]], and then using [[Cayley's theorem]] to embed it in the symmetric group on four letters.
 
==Eight-element subgroups (type (7))==
 
There are three of these subgroups:
 
<math>\{ (1,2,3,4), (1,3)(2,4), (1,4,3,2), (), (1,3), (2,4), (1,2)(3,4), (1,4)(3,2) \}</math>
 
<math>\{ (1,2,4,3), (1,4)(3,2), (1,3,4,2), (), (1,4), (3,2), (1,2)(3,4), (1,3)(2,4) \}</math>
 
<math>\{ (1,3,2,4), (1,2)(3,4), (1,4,2,3), (), (1,2), (3,4), (1,3)(2,4), (1,4)(2,3) \}</math>
 
===Subgroup properties satisfied by these subgroups===
 
* [[Sylow subgroup]]: These are all <math>2</math>-Sylow subgroups: their order is the largest power of <math>2</math> dividing the order of the group.
* [[Isomorph-conjugate subgroup]]: {{further|[[Sylow implies isomorph-conjugate]]}}
* [[Abnormal subgroup]]
* [[Self-normalizing subgroup]]
* [[Self-centralizing subgroup]]
* [[Procharacteristic subgroup]]
* [[Pronormal subgroup]]
* [[Contranormal subgroup]]
 
===Subgroup properties not satisfied by these subgroups===
 
* [[Core-free subgroup]]
 
===Effect of operators===
 
* [[Normal closure]]: The normal closure of any of these is the whole group.
* [[Normal core]]: The normal core of any of these is the Klein-four group, type (5).
* [[Normalizer]]: Each of these equals its normalizer.
* [[Centralizer]]: The centralizer of each of these equals its center, which is generated by the double transposition that is the square of its 4-cycle.
 
===Relation with other subgroups===
 
* Type (3) (double transposition): There is a bijection between the subgroups generated by double transpositions and the subgroups of order eight, given by either the centralizer or the normalizer operation.
* Type (4) (pair of disjoint transpositions): There is a bijection between the subgroups generated by disjoint pairs of transpositions and the dihedral groups, given by the normalizer operation.
* Type (6) (cyclic of order four): There is a bijection between the subgroups generated by 4-cycles and the subgroups of order eight, given by the normalizer operation.
 
==The three-element subgroup spanned by a 3-cycle (type (8))==
 
There are four such subgroups:
 
<math>\{ (), (1,2,3), (1,3,2) \}, \{ (), (1,2,4), (1,4,2) \}, \{ (), (2,3,4), (2,4,3) \}, \{ (), (1,3,4), (1,4,3) \}</math>
 
===Subgroup properties satisfied by these subgroups===
 
* [[Sylow subgroup]]: These are the 3-Sylow subgroups.
* [[Core-free subgroup]]
* [[Intermediately isomorph-conjugate subgroup]]
* [[NE-subgroup]]: Each such subgroup equals the intersection of its [[normalizer]] and [[normal closure]].
* [[Self-centralizing subgroup]]
 
===Subgroup properties not satisfied by these subgroups===
 
* [[Contranormal subgroup]]
* [[Self-normalizing subgroup]]
 
===Effect of operators===
 
* [[Normalizer]]: The normalizer is a symmetric group on the three letters moved by the 3-cycle (subgroup of type (9)). This establishes a bijection between the subgroups of types (8) and (9).
* [[Centralizer]]: This is the subgroup itself.
* [[Normal closure]]: The normal closure is the alternating group (type (10)).
* [[Normal core]]: The normal core is trivial.
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