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 n for a symmetric group on a set with n elements, which for convenience we take to be the set \{ 1,2, \dots, n \}.

This article contrasts the properties and behavior of symmetric groups of small degrees, specifically the symmetric groups of degree n for n = 0,1,2,3,4,5,6,7, compared with higher values. We shall use S_n to denote the symmetric group of degree n.

Order and basic information

Cycle decompositions and their relation to conjugacy class

Further information: Cycle decomposition for permutations, Understanding the cycle decomposition, Cycle decomposition theorem for permutations, Cycle type determines conjugacy class

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Even cycles and alternating group

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Table of order, conjugacy classes

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Some easy firsts involving orders

Looking at the table, we see the following:

  • n = 2 is the first case of a nontrivial symmetric group.
  • n = 3 is the first case of a non-abelian, as well as non-nilpotent symmetric group. All S_n, n \ge 3 are not nilpotent. In particular, there is no element whose order equals the order of the whole group.
  • n = 4 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 4! = 24, while the exponent is the lcm of 1,2,3,4, which is 12. All n \ge 4 have the property that the exponent of S_n is strictly less than its order.

Composition factors and simplicity

The initial cases

  • The cases n = 0,1 are unusual, in that S_n is trivial, so its composition series has length 0.
  • n = 2 is the only case where S_n is a simple group, since A_n is trivial and has index two.

The special cases of n = 3 and n = 4

  • n = 3 is the only case where S_n is a non-abelian metacyclic group (and hence also the only case of a non-abelian supersolvable group). S_3 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.
  • n = 4 is the only case where S_n is a solvable group that is not supersolvable. S_4 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. S_4 is also the only symmetric group whose chief series is not a composition series.

The cases n \ge 5

Further information: Alternating groups are simple

A_n is simple non-abelian for n \ge 5, so the composition series of S_n coincides with its unique chief series and has length two, with A_n being the intermediate subgroup. The composition factors are A_n and a cyclic group of order two.

Inner automorphisms and outer automorphisms; normal and characteristic subgroups

Completeness

Further information: Automorphism group of alternating group equals symmetric group, Symmetric groups are complete

General facts

For all n, the following are true about S_n and A_n. Note that since for n \ge 5, A_n is simple, most of these statements can be proved simply by checking them in all the small cases:

  • For all S_n and all A_n, every normal subgroup is characteristic.
  • Any two normal subgroups of S_n are comparable, so S_n is a normal-comparable group. The same is true for A_n.
  • With the exception of n = 4, every S_n and every A_n is a T-group: a normal subgroup of a normal subgroup is normal.
  • For every S_n and A_n except n = 4, there is a unique composition series that also equals the unique chief series.
  • For every S_n and A_n, 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.