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.

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.

Elements

Upto conjugacy

For convenience, we take the underlying set to be \{ 1,2,3,4,5 \}.

There are seven conjugacy classes, corresponding to the unordered integer partitions of 5 (for more information, refer cycle type determines conjugacy class). We use the notation of the cycle decomposition for permutations:

  1. 5 = 1 + 1 + 1 + 1 +1, i.e., five fixed points: The identity element. (1)
  2. 5 = 2 + 1 + 1 + 1, i.e., one 2-cycle and three fixed points: The transpositions, such as (1,2). (10)
  3. 5 = 3 + 1 + 1, i.e., one 3-cycle and two fixed points: The 3-cycles, such as (1,2,3). (20)
  4. 5 = 4 + 1, i.e., one 4-cycle and one fixed point: The 4-cycles, such as (1,2,3,4). (30)
  5. 5 = 5, i.e., one 5-cycle: The 5-cycles, such as (1,2,3,4,5). (24)
  6. 5 = 3 + 2: Permutations such as (1,2,3)(4,5). (20)
  7. 5 = 2 + 2 + 1: The double transpositions, such as (1,2)(3,4). (15)

Of these, types (1),(3),(5),(7) are conjugacy classes of even permutations, while types (2), (4), and (6) are conjugacy classes of odd permutations. The even permutations together form a subgroup, namely, the alternating group of degree five.

Upto automorphism

S_5 is a complete group: in particular, every automorphism of the group is inner. Thus, the equivalence classes under automorphisms are the same as the conjugacy classes.

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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