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The symmetric group is defined as the group of all permutations on a set of five elements, i.e., it is the symmetric group of degree five.
The symmetric group of degree five can be defined in the following equivalent ways:
|exponent||60||Elements of order .|
|derived length||--||not a solvable group.|
|nilpotency class||--||not a nilpotent group.|
|Frattini length||1||Frattini-free group: intersection of maximal subgroups is trivial.|
|minimum size of generating set||2|
|number of subgroups||156||--|
|number of conjugacy classes||7|
|number of conjugacy classes of subgroups||19|
COMPARE AND CONTRAST: Want to know more about how this group compares with symmetric groups of other degrees? Read contrasting symmetric groups of various degrees.
|Abelian group||No||, don't commute||is non-abelian, .|
|Nilpotent group||No||Centerless: The center is trivial||is non-nilpotent, .|
|Metacyclic group||No||No cyclic normal subgroup||is not metacyclic, .|
|Supersolvable group||No||No cyclic normal subgroup||is not supersolvable, .|
|Solvable group||No||The subgroup is simple non-abelian||is simple and hence not solvable, .|
|Complete group||Yes||Centerless and every automorphism's inner||Symmetric groups are complete except the ones of degree .|
|Monolithic group||Yes||Monolith is the alternating group||All symmetric groups are monolithic; is the only case the monolith is not the alternating group.|
|One-headed group||Yes||The alternating group is the unique maximal normal subgroup||True for all .|
For convenience, we take the underlying set to be .
There are seven conjugacy classes, corresponding to the unordered integer partitions of (for more information, refer cycle type determines conjugacy class). We use the notation of the cycle decomposition for permutations:
- , i.e., five fixed points: The identity element. (1)
- , i.e., one -cycle and three fixed points: The transpositions, such as . (10)
- , i.e., one -cycle and two fixed points: The -cycles, such as . (20)
- , i.e., one -cycle and one fixed point: The -cycles, such as . (30)
- , i.e., one -cycle: The -cycles, such as . (24)
- : Permutations such as . (20)
- : The double transpositions, such as . (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.
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.
Since is a complete group, it is isomorphic to its automorphism group, where each element of acts on by conjugation.
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
Further information: Subgroup structure of symmetric group:S5