> [!NOTE] Definition (Permutation of $n$ letters) > Let $n\in \mathbb{N}^{+}.$ A permutation of $n$ letters is a [[Permutation of a Set|permutation]] of $\mathbb{N}^{+}_{\leq n}$: that is a [[Bijection|bijection]] (or equivalently - in this case - an injection) on $\mathbb{N}^{+}_{\leq n}.$ **Notation**: permutation of $n$ letters are written using [[Two-Row Notation|two-row notation]] (that is a permutation is an ordered selection of $n$ letters without repetition as formalised below) or [[Cycle Notation|cycle notation]]. > [!NOTE] Definition 2 (Permutation of $n$ letters) > Let $n\in \mathbb{N}^{+}.$ Let $A$ be a [[Finite Set|finite set]] with [[Cardinality|cardinality]] $n.$ A permutation of $A$ is an [[Partial Permutation of n Letters (Ordered Selection)|n-permutation]] of $A$: that is, an [[Injection|injective]] [[List|list]] of length $n$ whose elements are taken from $A.$ > [!Example] > Contents # Properties **Algebra**: The set of these permutations, denoted $S_{n},$ form a group knowns as the [[Symmetric Groups of Finite Degree|nth degree symmetric group]]. By [[Number of Permutations of n Letters]], $S_{n}$ has cardinality $n!.$ **Cyclic Permutations**: A [[Cyclic Permutation of n Letters|k-cycle]] is a permutation with $i=\rho^{k-1}(i)$ for some $i$ that fixes each $j$ not in $\{ i,\rho(i),\rho^{2}(i),\dots,\rho^{k-1}(i) \}.$ **Cycle decomposition**: Note that [[Existence of Disjoint Cycle Decomposition for Permutations of n Letters|every n-permutation can be written uniquely as a product of disjoint cycles, up to the number of factors]]. Then, the [[Order of Product of Disjoint Permutations|order of the permutation]] is the lcm of the lengths of the cycles. The order with which we write the factors does not matter since [[Disjoint Permutations Commute|disjoint permutations]] commute. **Order**: [[Order of k-Cycle is k]]; [[Order of Product of Disjoint Permutations]]. **Parity:** Every permutation of $n$ letters can be expressed as a product of $2$-cycles, since [[k-cycles can be factored into 2-cycles|k-cycles can be factored into 2-cycles]]. The [[Parity of a Permutation of n letters|parity of the permutation]] is then defined as the parity of the number of $2$-cycles factors and is [[Parity of Permutation on n Letters is Well-Defined|well-defined]]. The [[Parity of the inverse of a permutation|parity of the inverse of a permutation]] is the same as the parity of the permutation. Note that the set of even permutations form a group known as the [[nth Alternating Group|alternating group]] which is a [[Alternating Group is a Normal Subgroup of Symmetric Group|normal subgroup]] of $S_{n}.$ **Derangement**: ...