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

Quote:

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 $|\text{Supp}(p)|\leqslant n$ , and let p be a permutation with $|\text{Supp}(p)|=n+1$ . Suppose that p moves i to j: $p (i)=j\ne i$ . 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 ... .