Difference between revisions of "Symmetric group:S5"

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Revision as of 00:25, 8 May 2008

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

Template:Not solvable

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.

Template:Complete

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.

Template:Jordan-uniqe

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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