Symmetric group:S4

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

Permutation definition

The symmetric group S_4 is defined as the group of all permutations on a set of 4 elements.

Presentation

Group properties

Solvability

This particular group is solvable

The commutator subgroup of S_4 is A_4, whose commutator subgroup is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.

Thus, S_4 is solvable of solvable length 3.

Nilpotence

This particular group is not nilpotent

Abelianness

This particular group is not Abelian

Simplicity

This particular group is not simple

Since S_4 has a nontrivial commutator subgroup, it is not Abelian.

Template:Not HN

There exist subnormal subgroups of S_4 which are not hypernormalized. For instance, the subgroup generated by the double transposition (12)(34) is2-subnormal (because it is normal in the subgroup generated by all double transpositions, which in turn is normal). However, it is not a hypernormalized subgroup, because its normalizer is a group of order 8 (a dihedral group) which is not normal.

Template:Complete

S_4 is a centerless group, and moreover, every automorphism of S_4 is inner. This can easily be checked by studying the effect of any automorphism oon the set of transpositions in S_4.

Endomorphisms

Automorphisms

Since S_4 is a complete group, it is isomorphic to its automorphism group, where each element of S_4 acts on S_4 by conjugation.

Endomorphisms

S_4 admits four kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these four):

  • The endomorphism to the trivial group
  • The identity map
  • The endomorphism to a group of order two, given by the sign homomorphism
  • The endomorphism to the symmetric group on 3 elements, with kernel being the Klein four-group

Retractions

All the four endomorphisms described above are retractions. Thus, in fact, every endomorphism of S_4 is equivalent, via some automorphism, to a retraction.

Subgroups

Normal subgroups

The only normal subgroups of S_4 are: the whole group, the trivial subgroup, A_4, and the Klein-four group.

Characteristic subgroups

Since every automorphism of S_4 is inner, the characteristic subgroups of S_4 are precisely the same as the normal subgroups.

Fully characteristic subgroups

In fact, all the normal subgroups described above are fully characteristic, a fact that can easily be checked from the explicit description of endomorphisms.

Retraction-invariant subgroups

Again, the retraction-invariant subgroups are precisely the same as the normal subgroups, as can be shown from the explicit description of retractions given above.

Subnormal subgroups

Apart from the normal subgroups, the only subnormal subgroups are the two-element subgroups corresponding to the double transpositions.

Permutable subgroups

The permutable subgroups are precisely the same as the normal subgroups.

Conjugate-permutable subgroups

The conjugate-permutable subgroups are precisely the same as the subnormal subgroups. In other words, apart from the normal subgroups, the two-element subgroups generated by double transpositions are also conjugate-permutable.

Hypernormalized subgroups

These are precisely the same as the normal subgroups.