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This article is about a particular group, viz a group unique upto isomorphism
This particular group is a finite group of order: 24
The symmetric group S4 is defined in the following equivalent ways:
- The group of all permutations, viz the symmetric group on a set of 4 elements
- The triangle group (not the von Dyck group, but its double) Δ(3,3,2)
- The group of all (not necessarily orientation-preserving) symmetries of the regular tetrahedron
- The group of orientation-preserving symmetries of the cube (or equivalently, the octahedron)
- The projective general linear group of order two over the field of three elements, viz: PGL(2,3)
This particular group is solvable
The commutator subgroup of S4 is A4, whose commutator subgroup is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.
Thus, S4 is solvable of solvable length 3.
This particular group is not nilpotent
This particular group is not Abelian
This particular group is not simple
Since S4 has a proper onntrivial commutator subgroup, it is not simple.
There exist subnormal subgroups of S4 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.
S4 is a centerless group, and moreover, every automorphism of S4 is inner. This can easily be checked by studying the effect of any automorphism oon the set of transpositions in S4.
Since S4 is a complete group, it is isomorphic to its automorphism group, where each element of S4 acts on S4 by conjugation.
S4 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
The endomorphisms of S4 are all retractions.
The only normal subgroups of S4 are: the whole group, the trivial subgroup, A4, and the Klein-four group.
In this group, the characteristic subgroups are the same as the normal subgroups. In other words, this is a N=C-group
This is because every automorphism of S4 is inner.
This can easily be checked from the explicit description of endomorphisms.
This can easily be checked from the explicit description of retractions.
Apart from the normal subgroups, the only subnormal subgroups are the two-element subgroups corresponding to the double transpositions.
The permutable subgroups are precisely the same as the normal 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.
The only transitively normal subgroups of S4 are the whole group, the trivial subgroup, and the alternating group. The Klein-four group is not transitively normal.
The only central factors are the whole group and the trivial subgroup.
The two proper abnormal subgroups (upto conjugacy) are: the symmetric group on three elemenst (which is of order 6), and the dihedral group on four elements (which is of order 8). These both occur as normalizers of Sylow subgroups (the one of order 8 is itself a Sylow subgroup, the other is the normalizer of a 3-cycle).
Apart from the two abnormal subgroups above, the two-element subgroup comprising a single transposition, is also a contranormal subgroup. So is the four-element subgroup comprising a single 4-cycle.
The subgroups that occur as retracts are: the whole group, the trivial subgroup, any two-element group for a single transposition, and any conjugate of the symmetric group on three elements.
The symmetric group S4 is contained in higher symmetric groups, most notably the symmetric group on five elements S5.
These include GL(2,3) whose inner automorphism group is S4 (specifically S4 is the quotient of GL(2,3) by its scalar matrices).
The symmetric group has five conjugacy classes, of which three are even and two are odd. The odd ones are the 4-cycle and the transposition. The even ones are the 3-cycle, the double transposition, and the identity element.
Implementation in GAP
The symmetric group on 4 elements is the 12th group of order 24, in GAP's small-group enumeration. So it can be described as: