# Changes

## Subgroup structure of symmetric group:S4

, 03:48, 18 February 2014
Table classifying subgroups up to automorphisms
{{quiz ad}}{{group-specific information|information type = subgroup structure|group = symmetric group:S4|connective = of}} The [[symmetric group:S4|symmetric group on of degree four letters]] has many sugbroupssubgroups.
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]].
# The trivial subgroup. Isomorphic to [[subgroup::trivial group]].(1)# The two-element subgroup generated by a transposition, such as $(1,2)$. Isomorphic to [[subgroup::cyclic group of order two]].(6)# The two-element subgroup generated by a double transposition, such as $(1,2)(3,4)$. Isomorphic to [[subgroup::cyclic group of order two]]. (3)# The four-element subgroup generated by two disjoint transpositions, such as $\langle (1,2) \ , \ (3,4) \rangle$. 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]]. ===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]] Tables for the prime $3$. ===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=quick information== * [[Direct factor]]* [[Central factor]]
==The Klein-four {{finite solvable group of double transpositions (type (5))==subgroup structure facts to check against}}
This <section begin="summary"/>===Quick summary==={| class="sortable" border="1"! Item !! Value|-| [[Number of subgroups]] || 30<br>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<br>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<br>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 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 a [[characteristic 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]]|-| [[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|-| [[maximal subgroup]]s || maximal subgroups have order 6 ([[S3 in S4]]), 8 ([[D8 in S4]]), and 12 ([[A4 in S4]]). |-| [[normal subgroup]]s || There are four normal subgroups: the whole group, the trivial subgroup, [[A4 in S4]], and [[normal V4 in S4]].|}
===Subgroup-defining functions yielding this subgroupTable classifying subgroups up to automorphisms===
* The [[Sylow-core]] for the prime $2$.* The second member of the [[derived series]].* The [[Fitting {{subgroup]].* The [[socle]]: it is in fact the unique [[minimal normal subgroup]].order sorting note}}
<small>{| class="sortable" border="1"! 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 (=Subgroup properties satisfied by this 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]] || $\{ (), (1,2)(3,4),$<br>$(1,3)(2,4), (1,4)(2,3) \}$ || [[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 !! -- !! -- !! --|}</small><section end="summary"/>
The given subgroup is a [[verbal subgroup]] on account ===Table classifying isomorphism types of being the second member of the derived series. Thus, it satisfies some subgroup properties, including:subgroups===
* {| class="sortable" border="1"! 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|-| [[Fully characteristic subgroupTrivial group]]|| 1 || 1 || 1 || 1 || 1 || 1|-| [[Cyclic group: It is invariant under any endomorphism of the whole Z2]] || 2 || 1 || 9 || 2 || 0 || 0|-| [[Cyclic group. {{further:Z3]] || 3 || 1 || 4 || 1 || 0 || 0|-|[[Verbal implies fully characteristicCyclic group:Z4]]}}|| 4 || 1 || 3 || 1 || 0 || 0|-* | [[ImageKlein four-closed fully characteristic subgroupgroup]]: Its image under any surjective homomorphism is fully characteristic in the image. {{further|| 4 || 2 || 4 || 2 || 1 || 1|-|[[Verbal implies image-closed fully characteristicSymmetric group:S3]]}}|| 6 || 1 || 4 || 1 || 0 || 0|-* | [[Image-closed characteristic subgroupDihedral group:D8]]: Its image under any surjective homomorphism is characteristic in the image. {{further|| 8 || 3 || 3 || 1 || 0 || 0|-|[[Verbal implies image-closed characteristicAlternating group:A4]]}}|| 12 || 3 || 1 || 1 || 1 || 1|-* | [[Characteristic subgroupSymmetric group:S4]]: It is characteristic in the whole group.|| 24 || 12 || 1 || 1 || 1 || 1|-! Total || -- || -- || 30 || 11 || 4 || 4|}
It also satisfies some other subgroup properties, such as:===Table listing number of subgroups by order===
* These numbers satisfy the [[Complemented normal subgroup]]* [[Self-centralizing subgroup]]* [[Coprime automorphism-faithful subgroup]]* [[Minimal normal subgroup]]* [[Fully normalized subgroupcongruence condition on number of subgroups of given prime power order]]: Every automorphism the number of it subgroups of order $p^r$ for a fixed nonnegative integer $r$ is realized as an inner automorphism in the whole groupcongruent to 1 mod $p$. In factFor $p = 2$, this means the whole group number is odd, and for $p = 3$, this means the [[holomorph]] of the subgroupnumber is congruent to 1 mod 3 (so it is among 1,4,7,...)
{| class="sortable" border==Subgroup properties not satisfied by this "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|}
* [[Direct factor]]* [[Central factor]]* [[Isomorph-free subgroup]]: Although the subgroup is characteristic, there exists an isomorphic subgroup. In fact, the ===Table listing numbers of subgroups of type (4) are all isomorphic to it.* [[Transitively normal subgroup]]by group property===
{| class="sortable" border=The two"1"! Group property !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup|-element subgroup generated by a transposition (type (|[[Cyclic group]] || 17 || 5 || 1 || 1|-|[[Abelian group]] || 21 || 7 || 2 || 2|-|[[Nilpotent group]] || 24 || 8 || 2 || 2))==|-|[[Solvable group]] || 30 || 11 || 4 || 4|}
===Subgroup properties satisfied Table listing numbers of subgroups by these subgroupssubgroup property===
* [[Retract]]: There is a retraction to any such {| class="sortable" border="1"! Subgroup property !! Occurences as subgroup !! Conjugacy classes of occurrences as subgroup !! Automorphism classes of occurrences as subgroup with kernel equal to the alternating group.* |-|[[Permutably complemented subgroupSubgroup]].|| 30 || 11 || 11* [[Core|-free subgroup]]* |[[Contranormal Normal subgroup]]|| 4 || 4 || 4* [[Conjugate|-permutable subgroup]]* |[[Automorph-permutable Characteristic subgroup]]|| 4 || 4 || 4|}
===Subgroup properties not satisfied by these subgroups=structure viewed as symmetric group==
* [[Pronormal subgroup]]* [[Polynormal subgroup]]* [[Permutable subgroup]]===Classification based on partition given by orbit sizes===
===Effect For any subgroup of operators===$S_4$, 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 $\{ 1,2 \}, \{ 3, 4 \}$|-| [[CentralizerA3 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 centralizer action is a [[transitive group action]], so only one orbit.|-| [[normal Kleinfour-four subgroup of S4]] || 1 || 4 || Yes || The action is a [[transitive groupaction]], so only one orbit.|-| [[non-normal Klein four-subgroups of type 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.* |-| [[NormalizerA4 in S4]]: || 1 || 4 || Yes || The normalizer action is a dihedral [[transitive group action]], so only one orbit.|-| whole group of order eight|| 1 || 4 || Yes || The action is a [[transitive group action]], of type (7)so only one orbit.|}