Symmetric Group¶

Definition¶

The symmetric group of degree $n$ , denoted $S_n$ , is the set of all permutations of an $n$ -element set, equipped with permutation composition.

Interpretation¶

$S_n$ collects every possible rearrangement of $n$ objects into one group, with composition as the operation: doing one rearrangement after another is again a rearrangement, doing nothing is the identity $e$ , and every rearrangement can be undone by its inverse permutation.

Six triangles with corners labelled 1, 2 and 3 in different orders: the identity and two rotations in the top row, and three reflections in the bottom row, the six elements of S3.

The figure shows the six elements of $S_3$ as the ways to relabel the corners of a triangle. The top row contains the identity $e$ and the two cycles $(1\,2\,3)$ and $(1\,3\,2)$ , which turn the triangle. The bottom row contains the three swaps $(2\,3)$ , $(1\,3)$ and $(1\,2)$ ; each keeps one label in place and exchanges the other two, like a reflection. The group operation is not commutative: composing right to left, $(1\,2)(1\,3)$ sends $1$ to $3$ , while $(1\,3)(1\,2)$ sends $1$ to $2$ . So $S_3$ is not an Abelian group, and it is the smallest group with this property.

The size $n!$ follows by counting: the image of $1$ can be chosen in $n$ ways, then the image of $2$ in $n-1$ ways, and so on, which gives the factorial $n(n-1)\cdots1$ $1) \dots 1$ . The same count of orderings appears in probability as the number of arrangements of $n$ distinct objects. For $n\le2$ the group is commutative. For $n\ge3$ it is not, since the two swaps above, acting only on the first three objects, still do not commute in $S_n$ .

Key Results¶

$S_n$ has $n!$ elements.
$(S_n,\circ)$ is a group.

Examples¶

$S_3$ contains all six permutations of a three-element set.
In $S_3$ , let $\sigma$ swap $1$ and $2$ , and let $\tau$ swap $2$ and $3$ . Applying the rightmost permutation first gives $(\sigma\circ\tau)(1)=2$ but $(\tau\circ\sigma)(1)=3$ $3$ . Thus these elements do not commute.

Prerequisites¶

Permutation Matrix
Abelian Group: $S_n$ is not Abelian when $n\ge3$ $3$ , as the example shows.
k-Permutation

Sources¶

Math II: Linear Algebra · Week 1

Aliases: symmetric group on n letters Keywords: permutation group Symbols: sigma, tau

Last updated: 20 October 2026 03:02