Added the sections permutations in set theory.

config/en/mkdocs.yaml

@@ -67,6 +67,7 @@ nav:
       - 'Sets': mathematics/set-theory/sets.md
       - 'Relations': mathematics/set-theory/relations.md
       - 'Maps': mathematics/set-theory/maps.md
+      - 'Permutations': mathematics/set-theory/permutations.md
     - 'Calculus': 
       - 'Limits': mathematics/calculus/limits.md
       - 'Continuity': mathematics/calculus/continuity.md

docs/en/mathematics/set-theory/permutations.md

@@ -0,0 +1,39 @@
+# Permutations
+
+## Definition
+
+> *Definition*: let $X$ be a set.
+>
+> * A bijection of $X$ to itself is called a permutation of $X$. The set of all permutations of $X$ is denoted by $\text{Sym}(X)$ and is called the symmetric group on $X$.
+> * The product $g \cdot h$ of two permutations $g,h$ in $\text{Sym}(X)$ is defined as the composition $g \circ h$ of $g$ and $h$.
+> * If $X = \{1, \dots, n\}$ we write $\mathrm{Sym}_n(X)$ instead of $\mathrm{Sym}(X)$.
+
+<br>
+
+> *Definition*: the identity map is defined as $\mathrm{id}: X \to X$ with $g = g \cdot \mathrm{id} = \mathrm{id} \cdot g$ for all $g$ in $\mathrm{Sym}(X)$. The inverse of $g$ denoted by $g^{-1}$ satisfies $g^{-1} \cdot g = g \cdot g^{-1} = \mathrm{id}$. 
+
+In matrix notation: let $g = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1\end{pmatrix}$ and $h = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3\end{pmatrix}$ with $g,h \in \mathrm{Sym}_3(X)$, then we can take
+
+$$
+    g \cdot h = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ \hline 2 & 1 & 3 \\ 3 & 2 & 1\end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1\end{pmatrix},
+$$
+
+and we have $g^{-1} = \begin{pmatrix} 2 & 3 & 1 \\1 & 2 & 3 \end{pmatrix}$. 
+
+<br>
+
+> *Theorem*: $\mathrm{Sym}_n$ has exactly $n!$ elements.
+
+??? note "*Proof*:"
+
+    A permutation can be described in a matrix notation by a $2$ by $n$ matrix with the numbers $1,\dots,n$ in the first row and the images in the second row. There are $n!$ possibilities to fill the second row.
+
+We can also omit the matrix notation and use the list notation for permutations then we have for $g = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1\end{pmatrix} = [2,3,1]$, as the first row speaks for itself.
+
+<br>
+
+> *Definition*: the order of a permutation $g$ is the smallest positive integer $m$ such that $g^m = \mathrm{id}$. 
+
+For example the order of the permutation $[2,1,3]$ in $\mathrm{Sym}_3$ is 2.
+
+If $g$ is a permutation in $\mathrm{Sym}_n$ then the permutations $g, g^2, g^3, \dots$ can not all be distinct, since there are only $n!$ distinct permutations in $\mathrm{Sym}_n$. So there must exists a $r < s$ such that $g^r = g^s$. Since $g$ is a bijection there must be $g^{s-r} = e$. So there exist positive numbers $m$ with $g^m = e$ and in particular a smallest such number. Therefore each permutation $g$ has a well-defined order.