Contrasting symmetric groups of various degrees
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The symmetric group on a set is defined as the group of all permutations on that set under composition. A bijection between two sets induces an isomorphism of the corresponding symmetric groups -- in particular, the isomorphism type of a symmetric group is completely determined by the cardinality of the set it acts on. Further, except for the case of sets of size zero and one, sets of distinct cardinalities have non-isomorphic symmetric groups. We shall use the term symmetric group of degree for a symmetric group on a set with elements, which for convenience we take to be the set .
This article contrasts the properties and behavior of symmetric groups of small degrees, specifically the symmetric groups of degree for , compared with higher values. We shall use to denote the symmetric group of degree .
Order and basic information
Cycle decompositions and their relation to conjugacy class[SHOW MORE]
Even cycles and alternating groupPLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Table of order, conjugacy classesPLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Some easy firsts involving orders
Looking at the table, we see the following:
- is the first case of a nontrivial symmetric group.
- is the first case of a non-abelian, as well as non-nilpotent symmetric group. All are not nilpotent. In particular, there is no element whose order equals the order of the whole group.
- is the first case of a symmetric group whose exponent is not equal to the order of the group. The order of the group is , while the exponent is the lcm of , which is . All have the property that the exponent of is strictly less than its order.
Composition factors and simplicity
The initial cases
- The cases are unusual, in that is trivial, so its composition series has length .
- is the only case where is a simple group, since is trivial and has index two.
The special cases of and
- is the only case where is a non-abelian metacyclic group (and hence also the only case of a non-abelian supersolvable group). has a cyclic normal subgroup (which is, in fact, a characteristic subgroup) of order three and a quotient of order two -- hence, it has a composition series of length two with both factors being cyclic groups.
- is the only case where is a solvable group that is not supersolvable. has a composition series of length four, with three composition factors being cyclic groups of order two and one composition factor being cyclic of order three. However, its unique chief series does not comprise only cyclic groups: the chief series has the Klein four-group of double transpositions and the alternating group. The chief factors are the Klein four-group, the cyclic group of order three, and the cyclic group of order two. is also the only symmetric group whose chief series is not a composition series.
Further information: Alternating groups are simple
is simple non-abelian for , so the composition series of coincides with its unique chief series and has length two, with being the intermediate subgroup. The composition factors are and a cyclic group of order two.
Inner automorphisms and outer automorphisms; normal and characteristic subgroups
Further information: Automorphism group of alternating group equals symmetric group, Symmetric groups are complete
- For , the group has the property that every automorphism is inner, but it is abelian (in fact, is, up to isomorphism, the only nontrivial group with a trivial automorphism group). Hence, it is not a complete group. Further information: Trivial automorphism group implies trivial or cyclic of order two
- For and , is the automorphism group of the group . Also, is a complete solvable group.
- For or , is the automorphism group of the simple non-abelian group . Hence, it is a complete group, and is also an almost simple group.
For all , the following are true about and . Note that since for , is simple, most of these statements can be proved simply by checking them in all the small cases:
- For all and all , every normal subgroup is characteristic.
- Any two normal subgroups of are comparable, so is a normal-comparable group. The same is true for .
- With the exception of , every and every is a T-group: a normal subgroup of a normal subgroup is normal.
- For every and except , there is a unique composition series that also equals the unique chief series.
- For every and , there is a unique chief series. Although there may be multiple composition series, the order of appearance of composition factors is the same in all.