Symmetric group

Definition

Let $X$ be a set. The symmetric group $\operatorname{Sym}(X)$ is the set of all bijections $\sigma \colon X \to X$, with composition of maps as the operation. For a positive integer $n$, the symmetric group $S_n$ is $\operatorname{Sym}(\{1, 2, \dots, n\})$. An element of $S_n$ is a permutation of degree $n$.

Remarks

The identity map is the identity element, and the inverse of $\sigma$ is the inverse map $\sigma^{-1}$. In this node, the product $\sigma \tau$ means first $\tau$, then $\sigma$. The group $S_n$ is not abelian for $n \geq 3$, because $(1\ 2)(2\ 3) \neq (2\ 3)(1\ 2)$. If $X$ and $Y$ have the same cardinality, then $\operatorname{Sym}(X)$ and $\operatorname{Sym}(Y)$ are isomorphic.

Explanations

L1 human

The symmetric group collects all the ways to shuffle the elements of a set. Two shuffles combine into one shuffle when you do them one after the other. Each shuffle can be undone, so the shuffles form a group.