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- 1 Definition
- 2 Group properties
- 3 Endomorphisms
- 4 Subgroups
The symmetric group is defined as the group of all permutations on a set of 4 elements.
This particular group is solvable
The commutator subgroup of is , whose commutator subgroup is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.
Thus, 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 has a nontrivial commutator subgroup, it is not Abelian.
There exist subnormal subgroups of which are not hypernormalized. For instance, the subgroup generated by the double transposition 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 (a dihedral group) which is not normal.
is a centerless group, and moreover, every automorphism of is inner. This can easily be checked by studying the effect of any automorphism oon the set of transpositions in .
Since is a complete group, it is isomorphic to its automorphism group, where each element of acts on by conjugation.
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
All the four endomorphisms described above are retractions. Thus, in fact, every endomorphism of is equivalent, via some automorphism, to a retraction.
The only normal subgroups of are: the whole group, the trivial subgroup, , and the Klein-four group.
Since every automorphism of is inner, the characteristic subgroups of are precisely the same as the normal 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.
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.
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.
These are precisely the same as the normal subgroups.