Group theory and generalizations Definition

Symmetric group

Builds on 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.

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Group concept
Symmetric group definition

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Tag	`0RJMNEN3BJ`
URI	`m2://0RJMNEN3BJ`
Aliases	`permutation-group-of-a-set`
Kind	definition
Writing	W1 stated
MSC	20B30
Defines	Symmetric group
Uses	Group

Symmetric group

Definition

Remarks

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