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 $S_{n},n\geq 3$ 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 $n\geq 4$ 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 $n\geq 5$

Further information: Alternating groups are simple

$A_{n}$ is simple non-abelian for $n\geq 5$, 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

• For $n=2$, the group $S_{2}$ has the property that every automorphism is inner, but it is abelian (in fact, $S_{2}$ 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 $n=3$ and $n=4$, $S_{n}$ is the automorphism group of the group $A_{n}$. Also, $S_{n}$ is a complete solvable group.
• For $n=5$ or $n\geq 7$, $S_{n}$ is the automorphism group of the simple non-abelian group $A_{n}$. Hence, it is a complete group, and is also an almost simple group.

General facts

For all $n$, the following are true about $S_{n}$ and $A_{n}$. Note that since for $n\geq 5$, $A_{n}$ 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.