# Thread: Show any permutation of a finite set may be expressed as a product of transpositions

1. ## Show any permutation of a finite set may be expressed as a product of transpositions

Prove any permutation of a finite set may be expressed as a product of transpositions by using induction on |Supp(p)|.
Here Supp(p) means the support of p, the elements which are moved by the map p.

2. Originally Posted by kevinlightman
Prove any permutation of a finite set may be expressed as a product of transpositions by using induction on |Supp(p)|.
Here Supp(p) means the support of p, the elements which are moved by the map p.
Base case |Supp(p)|=2 is rather obvious. Suppose the result holds for $\displaystyle |\text{Supp}(p)|\leqslant n$, and let p be a permutation with $\displaystyle |\text{Supp}(p)|=n+1$. Suppose that p moves i to j: $\displaystyle p (i)=j\ne i$. Then the product of the transposition (i j) with p fixes i and therefore moves at most n elements. So apply the inductive hypothesis to it ... .