Symmetric group

For a set $X$, the symmetric group $\mathrm{Sym}(X)$ is the group of all Bijection $X\to X$ under composition.

When $X=\{1,2,\dots ,n\}$, write $S_n=\mathrm{Sym}(\{1,\dots ,n\})$ - the symmetric group of degree $n$. Its elements are the Permutation of $\{1,\dots ,n\}$.

Order

$$|S_n|=n!.$$

Generators

$S_n$ is generated by:

• All transpositions $\{(ij)\}$ - a transposition is a 2-cycle.
• The two elements $(12)$ and $(12\cdots n)$ (a transposition and a long cycle).
• Adjacent transpositions $\{(ii+1):1\le i\le n-1\}$.

Cayley’s theorem

Every group $G$ embeds into $\mathrm{Sym}(G)$ via the regular action. So every finite group of order $n$ is a subgroup of $S_n$ (up to isomorphism).

Structure

• Centre: $Z(S_n)=\{e\}$ for $n\ge 3$. Trivial centre.
• Normal subgroups: for $n\ge 5$, the only normal subgroups of $S_n$ are $\{e\},A_n,S_n$ - i.e. $S_n/A_n\cong C_2$ is the only non-trivial quotient.
• Conjugacy classes: indexed by cycle shapes (= integer partitions of $n$). See Conjugacy.
• Signature $\sigma :S_n\to \{\pm 1\}$ is the unique non-trivial homomorphism to a smaller group; its kernel is $A_n$, the Alternating group.

Small Cases

• $S_1=\{e\}$, trivial.
• $S_2\cong C_2$, two elements.
• $S_3$, six elements: identity, three transpositions, two 3-cycles. Smallest non-abelian group.
• $S_4$, twenty-four elements; isomorphic to the rotation group of the cube.
• $S_5$, $120$ elements. Not solvable (Galois-theoretic obstruction to quintic formula).

Subgroups

Subgroups of $S_n$ are called permutation groups. By Cayley, every group is a permutation group.

Some standard examples:

• $A_n$ (alternating group), index 2.
• Cyclic subgroups $⟨\pi ⟩$ generated by individual permutations.
• Dihedral group $D_{2n}\le S_n$, the symmetries of an $n$-gon.
• Wreath products and direct products of smaller symmetric groups.