Symmetric group:S5

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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.

Group properties

The commutator subgroup of $S_{5}$ is $A_{5}$, which is simple and hence not solvable.

Nilpotence

This particular group is not nilpotent

Abelianness

This particular group is not Abelian

Simplicity

This particular group is not simple

Since $S_{5}$ has a proper nontrivial commutator subgroup, it is not simple.

$S_{5}$ 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 $S_{4}$.

The Jordan-Holder decomposition of $S_{5}$ is unique.

Endomorphisms

Automorphisms

Since $S_{5}$ is a complete group, it is isomorphic to its automorphism group, where each element of $S_{4}$ acts on $S_{4}$ by conjugation.

Endomorphisms

$S_{5}$ admits three kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these three):

• The endomorphism to the trivial group
• The identity map
• The endomorphism to a group of order two, given by the sign homomorphism

Retractions

The endomorphisms of $S_{5}$ are all retractions.

Subgroups

Normal subgroups

The only normal subgroups of $S_{5}$ are: the whole group, the trivial subgroup, and $A_{5}$ (the alternating group).

Characteristic subgroups

These are the same as the normal subgroups.

Fully characteristic subgroups

These are the same as the normal subgroups.

Retraction-invariant subgroups

These are the same as the normal subgroups.

Subnormal subgroups

These are the same as the normal subgroups. Thus $S_{5}$ is a T-group (that is, normality is transitive).

This follows from the fact that $A_{5}$ is a simple group.

Permutable subgroups

These are the same as the normal subgroups.

Conjugate-permutable subgroups

These are the same as the normal subgroups.

Contranormal subgroups

A subgroup is contranormal if and only if it is not contained within $A_{5}$.

Abnormal subgroups

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