Subgroup structure of symmetric group:S4

The symmetric group on four letters 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.

1. The trivial subgroup. Isomorphic to trivial group.(1)
2. The two-element subgroup generated by a transposition, such as $(1,2)$. Isomorphic to cyclic group of order two.(6)
3. The two-element subgroup generated by a double transposition, such as $(1,2)(3,4)$. Isomorphic to cyclic group of order two. (3)
4. The four-element subgroup generated by two disjoint transpositions, such as $⟨(1,2),(3,4)⟩$. Isomorphic to Klein four-group. (3)
5. The unique four-element subgroup comprising the identity and the three double transpositions. Isomorphic to Klein four-group. (1)
6. The four-element subgroup spanned by a 4-cycle. Isomorphic to cyclic group of order four.(3)
7. The eight-element subgroup spanned by a 4-cycle and a transposition that conjugates this cycle to its inverse. Isomorphic to dihedral group of order eight. This is also a 2-Sylow subgroup. (3)
8. The three-element subgroup spanned by a three-cycle. Isomorphic to cyclic group of order three.(4)
9. The six-element subgroup comprising all permutations that fix one element. Isomorphic to symmetric group on three elements. (4)
10. The alternating group: the subgroup of all even permutations. Isomorphic to alternating group:A4.(1)
11. 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 $\{1,2,3,4\}$ is given as:

$\{(),(1,2,3),(1,3,2),(1,2,4),(1,4,2),(2,3,4),(2,4,3),(1,3,4),(1,4,3),(1,3)(2,4),(1,4)(2,3),(1,2)(3,4)\}$

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 $3$.

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 information: Verbal implies fully characteristic
• Image-closed fully characteristic subgroup: Its image under any surjective homomorphism is fully characteristic in the image. Further information: Verbal implies image-closed fully characteristic
• Image-closed characteristic subgroup: Its image under any surjective homomorphism is characteristic in the image. Further information: 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 $\{1,2,3,4\}$, it is given as:

$\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$

Subgroup-defining functions yielding this subgroup

• 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.

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 information: Verbal implies fully characteristic
• Image-closed fully characteristic subgroup: Its image under any surjective homomorphism is fully characteristic in the image. Further information: Verbal implies image-closed fully characteristic
• Image-closed characteristic subgroup: Its image under any surjective homomorphism is characteristic in the image. Further information: 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 $2$-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 $\{1,2,3,4\}$, these subgroups are given by:

$\{(),(1,2)\},\{(),(2,3)\},\{(),(1,3)\},\{(),(1,4),\},\{(),(2,4)\}\{(),(3,4)\}$

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).
• Normalizer: The normalizer is a dihedral group of order eight, of type (7).

The two-element subgroup generated by a double transposition (type (3))

There are three such subgroups. In the symmetric group on $\{1,2,3,4\}$, these are given by:

$\{(),(1,2)(3,4)\},\{(1,3)(2,4)\},\{(1,4)(2,3)\}$.

These three subgroups are in bijection with the three 2-Sylow subgroups of order eight (type (7)) via the normalizer operation.

Subgroup properties satisfied by these subgroups

• Core-free subgroup
• Lattice-complemented subgroup: The symmetric group on any three elements forms a lattice complement to any of these subgroups.
• 2-subnormal subgroup: In fact, any such subgroup is normal in the Klein four-group $\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$, 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 information: 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 information: Conjugate-permutable not implies permutable
• 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 information: Pronormality is not intersection-closed

Subgroup properties not satisfied by these subgroups

• Permutably complemented subgroup
• 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

• 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 normalizer.

The non-normal Klein four-groups (type (4))

There are three such groups. These three groups are in bijection with the three 2-Sylow subgroups, via the normalizer operation.

$\{(),(1,2),(3,4),(1,2)(3,4)\},\{(),(1,3),(2,4),(1,3)(2,4)\},\{(),(1,4),(2,3),(1,4)(2,3)\}$

Subgroup properties satisfied by these subgroups

• Core-free subgroup
• Contranormal subgroup
• Self-centralizing subgroup
• Pronormal subgroup

Subgroup properties not satisfied by these subgroups

• Abnormal subgroup
• Self-normalizing subgroup
• 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:

$\{(),(1,2,3,4),(1,3)(2,4),(1,4,3,2)\},\{(),(1,2,4,3),(1,4)(3,2),(1,3,4,2)\},\{(),(1,3,2,4),(1,2)(3,4),(1,4,2,3)\}$.

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 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:

$\{(1,2,3,4),(1,3)(2,4),(1,4,3,2),(),(1,3),(2,4),(1,2)(3,4),(1,4)(3,2)\}$

$\{(1,2,4,3),(1,4)(3,2),(1,3,4,2),(),(1,4),(3,2),(1,2)(3,4),(1,3)(2,4)\}$

$\{(1,3,2,4),(1,2)(3,4),(1,4,2,3),(),(1,2),(3,4),(1,3)(2,4),(1,4)(2,3)\}$

Subgroup properties satisfied by these subgroups

• Sylow subgroup: These are all $2$-Sylow subgroups: their order is the largest power of $2$ dividing the order of the group.
• Isomorph-conjugate subgroup: Further information: Sylow implies isomorph-conjugate
• Abnormal subgroup
• Self-normalizing subgroup
• Self-centralizing subgroup
• Procharacteristic subgroup
• Pronormal subgroup
• Contranormal subgroup
• Permutably complemented 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:

$\{(),(1,2,3),(1,3,2)\},\{(),(1,2,4),(1,4,2)\},\{(),(2,3,4),(2,4,3)\},\{(),(1,3,4),(1,4,3)\}$

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
• Permutably complemented subgroup: The subgroups of type (7), i.e., the 2-Sylow subgroups, form permutable complements to these.

Subgroup properties not satisfied by these subgroups

• Contranormal subgroup
• Self-normalizing subgroup
• Retract

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.

The six-element subgroup comprising all permutations that fix one element (type (9))

There are four such subgroups, depending on which element we choose to fix.

$\{(),(1,2,3),(1,3,2),(1,2),(1,3),(2,3)\},\{(),(1,2,4),(1,4,2),(1,2),(1,4),(2,4)\},\{(),(1,3,4),(1,4,3),(1,3),(1,4),(3,4)\},\{(),(2,3,4),(2,4,3),(2,3),(2,4),(3,4)\}$.

Subgroup properties satisfied by these subgroups

• Maximal subgroup
• Abnormal subgroup
• Self-normalizing subgroup
• Self-centralizing subgroup
• Retract
• Isomorph-conjugate subgroup
• Intermediately isomorph-conjugate subgroup
• Contranormal subgroup