Given a finite set $X=\{a_{1},a_{2},\ldots,a_{n}\}$, a transposition is a permutation (bijective function of $X$ onto itself) $f$ such that there exist indices $i,j$ such that $f(a_{i})=a_{j}$, $f(a_{j})=a_{i}$ and $f(a_{k})=a_{k}$ for all other indices $k$. This is often denoted (in the cycle notation) as $(a,b)$.

Example: If $X=\{a,b,c,d,e\}$ the function $\sigma$ given by

is a transposition.

One of the main results on symmetric groups states that any permutation can be expressed as composition (product) of transpositions, and for any two decompositions of a given permutation, the number of transpositions is always even or always odd.