# Permutations

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 $S$.