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 is defined as the group of all permutations on a set of 4 elements.
Presentation
Group properties
The commutator subgroup of is , 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 has a proper nontrivial commutator subgroup, it is not simple.
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 .
The JordanHolder decomposition of is unique.
Endomorphisms
Automorphisms
Since is a complete group, it is isomorphic to its automorphism group, where each element of acts on by conjugation.
Endomorphisms
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 are all retractions.
Subgroups
Normal subgroups
The only normal subgroups of are: the whole group, the trivial subgroup, and (the alternating group).
Characteristic subgroups
These are the same as the normal subgroups.
Fully characteristic subgroups
These are the same as the normal subgroups.
Retractioninvariant subgroups
These are the same as the normal subgroups.
Subnormal subgroups
These are the same as the normal subgroups. Thus is a Tgroup (that is, normality is transitive).
This follows from the fact that is a simple group.
Permutable subgroups
These are the same as the normal subgroups.
Conjugatepermutable subgroups
These are the same as the normal subgroups.
Contranormal subgroups
A subgroup is contranormal if and only if it is not contained within .