Difference between revisions of "Subgroup structure of symmetric group:S4"

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The [[symmetric group:S4|symmetric group on four letters]] has many sugbroups.
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{{quiz ad}}
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{{group-specific information|
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information type = subgroup structure|
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group = symmetric group:S4|
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connective = of}}
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The [[symmetric group:S4|symmetric group of degree four]] has many subgroups.
  
 
Note that since <math>S_4</math> is a [[complete group]], every automorphism is inner, so the classification of subgroups upto conjugacy is equivalent to the classification of subgroups upto automorphism. In other words, every subgroup is an [[automorph-conjugate subgroup]].
 
Note that since <math>S_4</math> is a [[complete group]], every automorphism is inner, so the classification of subgroups upto conjugacy is equivalent to the classification of subgroups upto automorphism. In other words, every subgroup is an [[automorph-conjugate subgroup]].
  
# The trivial subgroup. Isomorphic to [[subgroup::trivial group]].(1)
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==Tables for quick information==
# 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 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 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]]. (3)
 
# The three-element subgroup spanned by a three-cycle. Isomorphic to [[subgroup::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)
 
# The whole group.(1)
 
 
 
==The alternating group (twelve-element characteristic subgroup) (type (10))==
 
 
 
This is a [[characteristic subgroup]]. The alternating group inside the symmetric group on the set <math>\{ 1,2,3,4 \}</math> is given as:
 
 
 
<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), (1,3)(2,4), (1,4)(2,3), (1,2)(3,4) \}</math>
 
 
 
===Subgroup-defining functions yielding this subgroup===
 
 
 
* The [[commutator subgroup]].
 
* The [[Jacobson radical]]: It is, in fact, the unique maximal normal subgroup.
 
* The subgroup generated by squares.
 
* The [[Sylow-closure]] for the prime <math>3</math>.
 
 
 
===Subgroup properties satisfied by this subgroup===
 
 
 
The alternating group is a [[verbal subgroup]] on account of being generated by commutators, or equivalently, on account of being generated by squares (actually, the two facts are closely related, and have to do with the fact that the symmetric group is a [[group generated by involutions]]). Thus, it satisfies some subgroup properties, including:
 
 
 
* [[Fully characteristic subgroup]]: It is invariant under any endomorphism of the whole group. {{further|[[Verbal implies fully characteristic]]}}
 
* [[Image-closed fully characteristic subgroup]]: Its image under any surjective homomorphism is fully characteristic in the image. {{further|[[Verbal implies image-closed fully characteristic]]}}
 
* [[Image-closed characteristic subgroup]]: Its image under any surjective homomorphism is characteristic in the image. {{further|[[Verbal implies image-closed characteristic]]}}
 
* [[Characteristic subgroup]]: It is characteristic in the whole group.
 
 
 
It also satisfies some other subgroup properties, such as:
 
 
 
* [[Complemented normal subgroup]]
 
* [[Order-unique subgroup]]
 
* [[Isomorph-free subgroup]]
 
* [[Self-centralizing subgroup]]
 
* [[Intermediately characteristic subgroup]]
 
* [[Transitively normal subgroup]]
 
 
 
===Subgroup properties not satisfied by this subgroup===
 
 
 
* [[Direct factor]]
 
* [[Central factor]]
 
 
 
==The Klein-four group of double transpositions (type (5))==
 
 
 
This is a [[characteristic subgroup]]. In the symmetric group on the set <math>\{ 1,2,3,4\}</math>, it is given as:
 
 
 
<math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math>
 
 
 
===Subgroup-defining functions yielding this subgroup===
 
 
 
* The [[Sylow-core]] for the prime <math>2</math>.
 
* The second member of the [[derived series]].
 
* The [[Fitting subgroup]].
 
* The [[socle]]: it is in fact the unique [[minimal normal subgroup]].
 
 
 
===Subgroup properties satisfied by this subgroup===
 
 
 
The given subgroup is a [[verbal subgroup]] on account of being the second member of the derived series. Thus, it satisfies some subgroup properties, including:
 
 
 
* [[Fully characteristic subgroup]]: It is invariant under any endomorphism of the whole group. {{further|[[Verbal implies fully characteristic]]}}
 
* [[Image-closed fully characteristic subgroup]]: Its image under any surjective homomorphism is fully characteristic in the image. {{further|[[Verbal implies image-closed fully characteristic]]}}
 
* [[Image-closed characteristic subgroup]]: Its image under any surjective homomorphism is characteristic in the image. {{further|[[Verbal implies image-closed characteristic]]}}
 
* [[Characteristic subgroup]]: It is characteristic in the whole group.
 
 
 
It also satisfies some other subgroup properties, such as:
 
 
 
* [[Complemented normal subgroup]]
 
* [[Self-centralizing subgroup]]
 
* [[Coprime automorphism-faithful subgroup]]
 
* [[Minimal normal subgroup]]
 
* [[Fully normalized subgroup]]: Every automorphism of it is realized as an inner automorphism in the whole group. In fact, the whole group is the [[holomorph]] of the subgroup.
 
* [[Normal-isomorph-free subgroup]]: There is no other normal subgroup of the whole group isomorphic to this  subgroup.
 
 
 
===Subgroup properties not satisfied by this subgroup===
 
 
 
* [[Direct factor]]
 
* [[Central factor]]
 
* [[Isomorph-free subgroup]]: Although the subgroup is characteristic, there exists an isomorphic subgroup. In fact, the subgroups of type (4) are all isomorphic to it.
 
* [[Transitively normal subgroup]]
 
* [[Intermediately characteristic subgroup]]: In fact, the subgroup is not characteristic inside any intermediate dihedral group, i.e., inside any of the <math>2</math>-Sylow subgroups.
 
 
 
===Other facts about this subgroup===
 
 
 
This subgroup can be obtained as follows. Consider the [[Klein-four group]], and use the embedding of this group inside the symmetric group on four letters given by [[Cayley's theorem]]. The image is precisely this subgroup.
 
 
 
==The two-element subgroup generated by a transposition (type (2))==
 
 
 
There are six such subgroups. In the symmetric group on <math>\{ 1,2,3,4\}</math>, these subgroups are given by:
 
 
 
<math>\{ (), (1,2) \}, \{ (), (2,3) \}, \{ (), (1,3) \}, \{ (), (1,4), \}, \{ (), (2,4) \} \{ (), (3,4) \}</math>
 
 
 
===Subgroup properties satisfied by these subgroups===
 
 
 
* [[Retract]]: There is a retraction to any such subgroup with kernel equal to the alternating group.
 
* [[Permutably complemented subgroup]]: This follows from being a retract.
 
* [[Core-free subgroup]]
 
* [[Contranormal subgroup]]
 
 
 
===Subgroup properties not satisfied by these subgroups===
 
 
 
* [[Pronormal subgroup]]
 
* [[Polynormal subgroup]]
 
* [[Permutable subgroup]]
 
* [[Conjugate-permutable subgroup]]
 
 
 
===Effect of operators===
 
  
* [[Centralizer]]: The centralizer is a Klein-four group, of type (4).
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{{finite solvable group subgroup structure facts to check against}}
* [[Normalizer]]: The normalizer is a dihedral group of order eight, of type (7).
 
  
==The two-element subgroup generated by a double transposition (type (3))==
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<section begin="summary"/>
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===Quick summary===
 +
{| class="sortable" border="1"
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! Item !! Value
 +
|-
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| [[Number of subgroups]] || 30<br>Compared with <math>S_n, n = 1,2,3,4,5,\dots</math>: 1,2,6,'''30''',156,1455,11300, 151221
 +
|-
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| [[Number of conjugacy classes of subgroups]] || 11<br>Compared with <math>S_n, n = 1,2,3,4,5,\dots</math>: 1,2,4,'''11''',19,56,96,296,554,1593
 +
|-
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| [[Number of automorphism classes of subgroups]] || 11<br>Compared with <math>S_n, n = 1,2,3,4,5,\dots</math>: 1,2,4,'''11''',19,37,96,296,554,1593
 +
|-
 +
| Isomorphism classes of [[Sylow subgroup]]s and the corresponding [[Sylow number]]s and [[fusion system]]s || 2-Sylow: [[dihedral group:D8]] (order 8), Sylow number is 3, fusion system is [[non-inner non-simple fusion system for dihedral group:D8]]<br>3-Sylow: [[cyclic group:Z3]], Sylow number is 4, fusion system is [[non-inner fusion system for cyclic group:Z3]]
 +
|-
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| [[Hall subgroup]]s || Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups
 +
|-
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| [[maximal subgroup]]s || maximal subgroups have order 6 ([[S3 in S4]]), 8 ([[D8 in S4]]), and 12 ([[A4 in S4]]).
 +
|-
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| [[normal subgroup]]s || There are four normal subgroups: the whole group, the trivial subgroup, [[A4 in S4]], and [[normal V4 in S4]].
 +
|}
  
There are three such subgroups. In the symmetric group on <math>\{ 1,2,3,4\}</math>, these are given by:
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===Table classifying subgroups up to automorphisms===
  
<math>\{ (), (1,2)(3,4) \}, \{ (1,3)(2,4) \}, \{ (1,4)(2,3) \}</math>.
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{{subgroup order sorting note}}
  
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.
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<small>
 +
{| class="sortable" border="1"
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! Automorphism class of subgroups !! Representative !! Isomorphism class !! [[Order of a group|Order]] of subgroups !! [[Index of a subgroup|Index]] of subgroups !! Number of conjugacy classes (=1 iff [[automorph-conjugate subgroup]]) !! Size of each conjugacy class (=1 iff [[normal subgroup]]) !! Number of subgroups (=1 iff [[characteristic subgroup]])!! Isomorphism class of quotient (if exists) !! [[Subnormal depth]] (if subnormal) !! Note
 +
|-
 +
| trivial subgroup || <math>\{ () \}</math> || [[trivial group]] || 1 || 24 || 1 || 1 || 1 || [[symmetric group:S4]] || 1 ||
 +
|-
 +
| [[S2 in S4]] || <math>\{ (), (1,2) \}</math> || [[cyclic group:Z2]] || 2 || 12 || 1 || 6 || 6 || -- || -- ||
 +
|-
 +
| [[subgroup generated by double transposition in S4]] || <math>\{ (), (1,2)(3,4) \}</math> || [[cyclic group:Z2]] || 2 || 12 || 1 || 3 || 3 || -- || 2 ||
 +
|-
 +
| [[Z4 in S4]] || <math>\langle (1,2,3,4) \rangle</math> || [[cyclic group:Z4]] || 4 || 6 ||  1 || 3 || 3 || -- || -- ||
 +
|-
 +
| [[normal Klein four-subgroup of S4]] || <math>\{ (), (1,2)(3,4), </math><br><math>(1,3)(2,4), (1,4)(2,3) \}</math> || [[Klein four-group]] || 4 || 6 || 1 || 1 || 1 || [[symmetric group:S3]] || 1 || 2-core
 +
|-
 +
| [[non-normal Klein four-subgroups of S4]] || <math>\langle (1,2), (3,4) \rangle</math> || [[Klein four-group]] || 4 || 6 || 1 || 3 || 3 || -- || -- ||
 +
|-
 +
| [[D8 in S4]] || <math>\langle (1,2,3,4), (1,3) \rangle</math> || [[dihedral group:D8]] || 8 || 3 || 1 || 3 || 3 || -- || -- || 2-Sylow, fusion system is [[non-inner non-simple fusion system for dihedral group:D8]]
 +
|-
 +
| [[A3 in S4]] || <math>\{ (), (1,2,3), (1,3,2) \}</math>  || [[cyclic group:Z3]] || 3 || 8 || 1 || 4 || 4 || -- || -- || 3-Sylow, fusion system is [[non-inner fusion system for cyclic group:Z3]]
 +
|-
 +
| [[S3 in S4]] || <math>\langle (1,2,3), (1,2) \rangle</math> ||[[symmetric group:S3]] || 6 || 4 || 1 || 4 || 4 || -- || -- ||
 +
|-
 +
| [[A4 in S4]] || <math>\langle (1,2,3), (1,2)(3,4) \rangle</math> || [[alternating group:A4]] || 12 || 2 || 1 || 1 || 1 || [[cyclic group:Z2]] || 1 ||
 +
|-
 +
| whole group || <math>\langle (1,2,3,4), (1,2) \rangle</math> || [[symmetric group:S4]] || 24 || 1 || 1 || 1 || 1 || [[trivial group]] || 0 ||
 +
|-
 +
! Total (11 rows) !! -- !! -- !! -- !! -- !! 11 !! -- !! 30 !! -- !! -- !! --
 +
|}
 +
</small>
 +
<section end="summary"/>
  
===Subgroup properties satisfied by these subgroups===
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===Table classifying isomorphism types of subgroups===
  
* [[Core-free subgroup]]
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{| class="sortable" border="1"
* [[Lattice-complemented subgroup]]: The symmetric group on any three elements forms a lattice complement to any of these subgroups.
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! Group name !! Order !! Second part of GAP ID (first part is order) !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
* [[2-subnormal subgroup]]: In fact, any such subgroup is normal in the Klein-four group <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)\}</math>, which is normal in the whole group. More strongly, it is a [[subgroup of Abelian normal subgroup]]. In fact, it is an example of a 2-subnormal subgroup whose normalizer is an abnormal subgroup. {{further|[[2-subnormal and abnormal normalizer not implies normal]]}}
+
|-
* [[Conjugate-permutable subgroup]]: In fact, this is an example of a conjugate-permutable subgroup that is not permutable. {{further|[[Conjugate-permutable not implies permutable]]}}
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| [[Trivial group]] || 1 || 1 || 1 || 1 || 1 || 1
* [[Automorph-permutable subgroup]]
+
|-
* [[Intersection of pronormal subgroups]]: Any of these subgroups can be obtained as the intersection of the normal Klein-four group (type (5)) and a non-normal Klein-four group (type (4)). Both of these are pronormal in the whole group, and thus, the given subgroup is an intersection of pronormal subgroups. {{further|[[Pronormality is not intersection-closed]]}}
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| [[Cyclic group:Z2]] || 2 || 1 || 9 || 2 || 0 || 0
 +
|-
 +
| [[Cyclic group:Z3]] || 3 || 1 || 4 || 1 || 0 || 0
 +
|-
 +
| [[Cyclic group:Z4]] || 4 || 1 || 3 || 1 || 0 || 0
 +
|-
 +
| [[Klein four-group]] || 4 || 2 || 4 || 2 || 1 || 1
 +
|-
 +
| [[Symmetric group:S3]] || 6 || 1 || 4 || 1 || 0 || 0
 +
|-
 +
| [[Dihedral group:D8]] || 8 || 3 || 3 || 1 || 0 || 0
 +
|-
 +
| [[Alternating group:A4]] || 12 || 3 || 1 || 1 || 1 || 1
 +
|-
 +
| [[Symmetric group:S4]] || 24 || 12 || 1 || 1 || 1 || 1
 +
|-
 +
! Total || -- || -- || 30 || 11 || 4 || 4
 +
|}
  
===Subgroup properties not satisfied by these subgroups===
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===Table listing number of subgroups by order===
  
* [[Permutably complemented subgroup]]
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These numbers satisfy the [[congruence condition on number of subgroups of given prime power order]]: the number of subgroups of order <math>p^r</math> for a fixed nonnegative integer <math>r</math> is congruent to 1 mod <math>p</math>. For <math>p = 2</math>, this means the number is odd, and for <math>p = 3</math>, this means the number is congruent to 1 mod 3 (so it is among 1,4,7,...)
* [[Contranormal subgroup]]
 
* [[Retract]]
 
* [[2-hypernormalized subgroup]]: The normalizer of this subgroup is a dihedral group of order eight, which is not normal in the whole group.
 
* [[Permutable subgroup]]
 
* [[Pronormal subgroup]]
 
  
===Effect of operators===
+
{| class="sortable" border="1"
 +
! Group order !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
 +
|-
 +
| 1 || 1 || 1 || 1 || 1
 +
|-
 +
| 2 || 9 || 2 || 0 || 0
 +
|-
 +
| 3 || 4 || 1 || 0 || 0
 +
|-
 +
| 4 || 7 || 3 || 1 || 1
 +
|-
 +
| 6 || 4 || 1 || 0 || 0
 +
|-
 +
| 8 || 3 || 1 || 0 || 0
 +
|-
 +
| 12 || 1 || 1 || 1 || 1
 +
|-
 +
| 24 || 1 || 1 || 1 || 1
 +
|-
 +
! Total || 30 || 11 || 4 || 4
 +
|}
  
* [[Normal closure]]: The normal closure is the Klein-four group, type (5).
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===Table listing numbers of subgroups by group property===
* [[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 closure.
 
  
==The non-normal Klein-four groups (type (4))==
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{| class="sortable" border="1"
 +
! Group property !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
 +
|-
 +
|[[Cyclic group]] || 17 || 5 || 1 || 1
 +
|-
 +
|[[Abelian group]] || 21 || 7 || 2 || 2
 +
|-
 +
|[[Nilpotent group]] || 24 || 8 || 2 || 2
 +
|-
 +
|[[Solvable group]] || 30 || 11 || 4 || 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.
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===Table listing numbers of subgroups by subgroup property===
  
<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>
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{| class="sortable" border="1"
 +
! Subgroup property !! Occurences as subgroup !! Conjugacy classes of occurrences as subgroup !! Automorphism classes of occurrences as subgroup
 +
|-
 +
|[[Subgroup]] || 30 || 11 || 11
 +
|-
 +
|[[Normal subgroup]] || 4 || 4 || 4
 +
|-
 +
|[[Characteristic subgroup]] || 4 || 4 || 4
 +
|}
  
===Subgroup properties satisfied by these subgroups===
+
==Subgroup structure viewed as symmetric group==
  
* [[Core-free subgroup]]
+
===Classification based on partition given by orbit sizes===
* [[Contranormal subgroup]]
 
* [[Self-centralizing subgroup]]
 
* [[Pronormal subgroup]]
 
  
===Subgroup properties not satisfied by these subgroups===
+
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.
  
* [[Abnormal subgroup]]
+
{| class="sortable" border="1"
* [[Self-normalizing subgroup]]
+
! Conjugacy class of subgroups !! Size of conjugacy class !! Induced partition of 4 !! Direct product of transitive subgroups on each orbit? !! Illustration with representative
* [[Permutable subgroup]]
+
|-
* [[Conjugate-permutable subgroup]]
+
| 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.
 +
|}

Latest revision as of 03:48, 18 February 2014

TAKE A QUIZ ON THIS TOPIC and test the quality of your understanding of it
This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S4.
View subgroup structure of particular groups | View other specific information about symmetric group:S4

The symmetric group of degree four has many subgroups.

Note that since S_4 is a complete group, every automorphism is inner, so the classification of subgroups upto conjugacy is equivalent to the classification of subgroups upto automorphism. In other words, every subgroup is an automorph-conjugate subgroup.

Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group


Quick summary

Item Value
Number of subgroups 30
Compared with S_n, n = 1,2,3,4,5,\dots: 1,2,6,30,156,1455,11300, 151221
Number of conjugacy classes of subgroups 11
Compared with S_n, n = 1,2,3,4,5,\dots: 1,2,4,11,19,56,96,296,554,1593
Number of automorphism classes of subgroups 11
Compared with S_n, n = 1,2,3,4,5,\dots: 1,2,4,11,19,37,96,296,554,1593
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems 2-Sylow: dihedral group:D8 (order 8), Sylow number is 3, fusion system is non-inner non-simple fusion system for dihedral group:D8
3-Sylow: cyclic group:Z3, Sylow number is 4, fusion system is non-inner fusion system for cyclic group:Z3
Hall subgroups Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups
maximal subgroups maximal subgroups have order 6 (S3 in S4), 8 (D8 in S4), and 12 (A4 in S4).
normal subgroups There are four normal subgroups: the whole group, the trivial subgroup, A4 in S4, and normal V4 in S4.

Table classifying subgroups up to automorphisms

TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.

Automorphism class of subgroups Representative Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes (=1 iff automorph-conjugate subgroup) Size of each conjugacy class (=1 iff normal subgroup) Number of subgroups (=1 iff characteristic subgroup) Isomorphism class of quotient (if exists) Subnormal depth (if subnormal) Note
trivial subgroup \{ () \} trivial group 1 24 1 1 1 symmetric group:S4 1
S2 in S4 \{ (), (1,2) \} cyclic group:Z2 2 12 1 6 6 -- --
subgroup generated by double transposition in S4 \{ (), (1,2)(3,4) \} cyclic group:Z2 2 12 1 3 3 -- 2
Z4 in S4 \langle (1,2,3,4) \rangle cyclic group:Z4 4 6 1 3 3 -- --
normal Klein four-subgroup of S4 \{ (), (1,2)(3,4),
(1,3)(2,4), (1,4)(2,3) \}
Klein four-group 4 6 1 1 1 symmetric group:S3 1 2-core
non-normal Klein four-subgroups of S4 \langle (1,2), (3,4) \rangle Klein four-group 4 6 1 3 3 -- --
D8 in S4 \langle (1,2,3,4), (1,3) \rangle dihedral group:D8 8 3 1 3 3 -- -- 2-Sylow, fusion system is non-inner non-simple fusion system for dihedral group:D8
A3 in S4 \{ (), (1,2,3), (1,3,2) \} cyclic group:Z3 3 8 1 4 4 -- -- 3-Sylow, fusion system is non-inner fusion system for cyclic group:Z3
S3 in S4 \langle (1,2,3), (1,2) \rangle symmetric group:S3 6 4 1 4 4 -- --
A4 in S4 \langle (1,2,3), (1,2)(3,4) \rangle alternating group:A4 12 2 1 1 1 cyclic group:Z2 1
whole group \langle (1,2,3,4), (1,2) \rangle symmetric group:S4 24 1 1 1 1 trivial group 0
Total (11 rows) -- -- -- -- 11 -- 30 -- -- --


Table classifying isomorphism types of subgroups

Group name Order Second part of GAP ID (first part is order) Occurrences as subgroup Conjugacy classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
Trivial group 1 1 1 1 1 1
Cyclic group:Z2 2 1 9 2 0 0
Cyclic group:Z3 3 1 4 1 0 0
Cyclic group:Z4 4 1 3 1 0 0
Klein four-group 4 2 4 2 1 1
Symmetric group:S3 6 1 4 1 0 0
Dihedral group:D8 8 3 3 1 0 0
Alternating group:A4 12 3 1 1 1 1
Symmetric group:S4 24 12 1 1 1 1
Total -- -- 30 11 4 4

Table listing number of subgroups by order

These numbers satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order p^r for a fixed nonnegative integer r is congruent to 1 mod p. For p = 2, this means the number is odd, and for p = 3, this means the number is congruent to 1 mod 3 (so it is among 1,4,7,...)

Group order Occurrences as subgroup Conjugacy classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
1 1 1 1 1
2 9 2 0 0
3 4 1 0 0
4 7 3 1 1
6 4 1 0 0
8 3 1 0 0
12 1 1 1 1
24 1 1 1 1
Total 30 11 4 4

Table listing numbers of subgroups by group property

Group property Occurrences as subgroup Conjugacy classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
Cyclic group 17 5 1 1
Abelian group 21 7 2 2
Nilpotent group 24 8 2 2
Solvable group 30 11 4 4

Table listing numbers of subgroups by subgroup property

Subgroup property Occurences as subgroup Conjugacy classes of occurrences as subgroup Automorphism classes of occurrences as subgroup
Subgroup 30 11 11
Normal subgroup 4 4 4
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.

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.