Finite permutation/Representation with transpositions/Fact/Proof

Proof

We proof the statement by induction over the cardinality of the set ${}M$. For ${}{\#\left(M\right)}=1$, there is nothing to show, so let ${}{\#\left(M\right)}\geq 2$. The identity is the empty product of transpositions. So suppose that ${}\pi$ is not the identity, and let ${}\pi (x)=y\neq x$. Let ${}\tau$ be the transposition which swaps ${}x$ and ${}y$. Then ${}y$ is a fixed point of ${}\pi \tau$, and we can consider ${}\pi \tau$ as a permutation on ${}M'=M\setminus \{y\}$. By the induction hypothesis, there exist transpositions ${}\tau _{j}$ on ${}M'$ such that ${}\pi \tau =\prod _{j}\tau _{j}$ on ${}M'$. This does also hold on ${}M$, and we get ${}\pi ={\left(\prod _{j}\tau _{j}\right)}\tau$.

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