Subgroup structure of symmetric group:S4

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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

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:

It also satisfies some other subgroup properties, such as:

Subgroup properties not satisfied by this subgroup

The Klein-four group of double transpositions (type (5))

This is a characteristic subgroup.

Subgroup-defining functions yielding this 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:

It also satisfies some other subgroup properties, such as:

Subgroup properties not satisfied by this subgroup

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

Subgroup properties satisfied by these subgroups

Subgroup properties not satisfied by these subgroups

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).