Equality of Two Permutations

Two permutations f and g of degree n are said to be equal if we have f\left( a \right) = g\left( a \right), \forall a \in S.


If f = \left(  {\begin{array}{*{20}{c}}1&2&3&4 \\ 2&3&4&1 \end{array}}  \right) and g = \left(  {\begin{array}{*{20}{c}}1&4&1&3 \\ 3&1&2&4 \end{array}}  \right) are two permutations of degree 4, then we havef = g. Here we see that both f and g replace 1 by 2, 2 by 3, 3 by 4, and 4 by 1.

If 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) is a permutation of degree n, we can write it in several ways. The interchange of columns will not change the permutation. Thus, we can write

If f = \left(  {\begin{array}{*{20}{c}}{{a_2}}&{{a_3}}&{{a_1}}& \cdots  &{{a_n}} \\ {{b_2}}&{{b_3}}&{{b_1}}& \cdots &{{b_n}} \end{array}}  \right) = \left( {\begin{array}{*{20}{c}}{{a_n}}&{{a_1}}& \cdots  &{{a_n} - 1} \\ {{b_n}}&{{b_1}}& \cdots &{{b_n} - 1} \end{array}}  \right) then

f = \left(  {\begin{array}{*{20}{c}}{{a_n} - 1}&{{a_n}}&{{a_n} - 2}& \cdots  &{{a_1}} \\ {{b_n} - 1}&{{b_n}}&{{b_n} - 2}& \cdots  &{{b_1}} \end{array}} \right)

Therefore, if f and g are two permutations of the same elements of degree n, then it is always possible to write g in such a way that the first row of g coincide with the second row of f.

Total Number of Distinct Permutations of Degree n:

If S is a finite set having n distinct elements, then we shall have n{!} distinct arrangements of the elements of S. Therefore there will be n{!} distinct permutations of degree n. If{P_n} be the set consisting of all permutations of degree n. If {P_n} be the set containing of all permutations of degree n then the set {P_n} will have n{!} distinct elements.

This set {P_n} is called the symmetric set of permutations of degree n. Sometimes it is also denoted by {S_n}.

Thus, {P_n} = (f:f is a permutation of degree n).

The set{P_3} of all permutation of degree 3 will have 3!, i.e., 6 elements. Obviously

 {P_3} = \left\{ {\left(  {\begin{array}{*{20}{c}}1&2&3 \\ 1&2&3 \end{array}} \right), \left(  {\begin{array}{*{20}{c}}1&2&3 \\ 3&1&2 \end{array}}  \right),\left( {\begin{array}{*{20}{c}}1&2&3 \\ 2&3&1 \end{array}}  \right),\left( {\begin{array}{*{20}{c}}1&2&3 \\ 3&2&1  \end{array}} \right),\left( {\begin{array}{*{20}{c}}1&2&3 \\ 1&3&2  \end{array}} \right),\left( {\begin{array}{*{20}{c}}1&2&3 \\ 2&1&3  \end{array}} \right)} \right\}