> [!NOTE] **Definition** (Symmetric group of a set) > Suppose $A$ is a set. The symmetric group (or *permutation group*) of $A$, denoted $\text{Sym}(A)$, is the set of [[Permutation of a Set|permutations]] of $A$ under [[Function Composition|function composition]]. # Properties By [[Symmetric Group is Group]], $\text{Sym}(A)$ is indeed a group under function composition. **Isomorphism between symmetric groups**: The [[Symmetric Groups of Finite Degree|symmetric group of degree n]] (or $n$th symmetric group or $S_{n}$) is defined as $\text{Sym}(\{ k\in\mathbb{N}^{+} \mid k \leq n \}).$ Note that the [[Symmetric Groups of Same Order Are Isomorphic|symmetric groups of finite sets of equal cardinality are isomorphic]]. As a consequence, results can be proved about the $n$th *symmetric group* which then apply to all symmetric groups on *finite sets* with $n$ elements.