Suppose S is a finite set having n distinct elements. Then a one-one mapping of S onto itself is called a permutation of degree n. The number of elements in the finite set S is known as the degree of permutation.


Symbol for a Permutation

Let S = \left\{ {{a_1},{a_2},{a_3}, \ldots ,{a_n}} \right\} be a finite set having n distinct elements. If f:S \to S in one-one mapping, then f is a permutation of degree n.

Let f\left( {{a_1}} \right) = {b_1}, f\left( {{a_2}} \right) = {b_2}, f\left( {{a_3}} \right) = {b_3}, …, f\left( {{a_n}} \right) = {b_n} where \left\{ {{b_1},{b_2},{b_3}, \ldots ,{b_n}} \right\} = \left\{ {{a_1},{a_2},{a_3}, \ldots ,{a_n}} \right\} i.e. {b_1},{b_2},{b_3}, \ldots ,{b_n} is one arrangement of the n elements {a_2}, \ldots ,{a_n}, \ldots .

It is customary to write a permutation in a two line symbol. In this notation we write:

 f = \left( {\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}}& \cdots &{{a_n}} \\ {{b_1}}&{{b_2}}&{{b_3}}& \cdots &{{b_n}} \end{array}} \right)

i.e. each element in the second row is the f - image of the element of the first row lying directly above it.

If S = \left\{ {1,2,3,4} \right\} is a finite set of order four then,

 f = \left( {\begin{array}{*{20}{c}} 1&2&3&4 \\ 2&4&1&3 \end{array}} \right)

 g = \left( {\begin{array}{*{20}{c}} 1&2&3&4 \\ 1&3&2&4 \end{array}} \right)

 h = \left( {\begin{array}{*{20}{c}} 1&2&3&4 \\ 1&2&4&3 \end{array}} \right)

etc., are all permutations of degree four. Here in the permutation f the elements 1, 2, 3, 4 are replaced respectively by the elements 2, 4, 1, 3. Thus f\left( 1 \right) = 2,f\left( 2 \right) = 4, f\left( 3 \right) = 1,f\left( 4 \right) = 3. Similarly, g\left( 1 \right) = 1,g\left( 2 \right) = 3, g\left( 3 \right) = 2,g\left( 4 \right) = 4 and h\left( 1 \right) = 1,h\left( 2 \right) = 2, h\left( 3 \right) = 4,h\left( 4 \right) = 3.

Thus for a permutation f on S, we just put the elements of S in one row in any order we like and below each element of this row we put down its image under, f,g or h to obtain another row of elements of S.