Matrix Representation of the Permutation Group

The set of all permutations of forms a group under the multiplication (composition) of permutations; that is, it meets the requirements of closure, existence of identity and inverses, and associativity. We can set up a bijection between and a set of binary matrices (the permutation matrices) that preserves this structure under the operation of matrix multiplication. The bijection associates the permutation with the matrix , with zeros everywhere except for ones at row, column , for .