Subgroup structure of symmetric group:S4

The symmetric group on four letters has many sugbroups.

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 $\langle (1,2) \ , \ (3,4) \rangle$. 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. (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.

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.

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.

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

The two-element subgroup generated by a transposition (type (2))

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.
• Core-free subgroup
• Contranormal subgroup
• Conjugate-permutable subgroup
• Automorph-permutable subgroup

Subgroup properties not satisfied by these subgroups

• Pronormal subgroup
• Polynormal subgroup
• 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).