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Order of Permutation Group

  • Last Updated : 15 Mar, 2021

Order of Permutation-: For a given permutation P if  Pn= I (identity permutation) , then n is the order of permutation.

Let a permutation P=\begin{pmatrix} a & b & c\\ d & d & e \end{pmatrix}

and  Pn = I = \begin{pmatrix} a & b & c\\ a & b & c \end{pmatrix}

Then n is the order of permutation.

Example 1-:  How many times \begin{pmatrix} 1 & 2 & 3&4\\ 1 & 3 & 4&2 \end{pmatrix}   be multiplied to itself to produce \begin{pmatrix} 1 & 2 & 3&4\\ 1 & 2 & 3&4 \end{pmatrix}



Solution-: Let      P=\begin{pmatrix} 1 & 2 & 3&4\\ 1 & 3 & 4&2 \end{pmatrix}

                   Then P2=P.P=\begin{pmatrix} 1 & 2 & 3&4\\ 1 & 3 & 4&2 \end{pmatrix}\begin{pmatrix} 1 & 2 & 3&4\\ 1 & 3 & 4&2 \end{pmatrix}

                             P2=\begin{pmatrix} 1 & 2 & 3&4\\ 1 & 4 & 2&3 \end{pmatrix}

                             P3= P2.P=\begin{pmatrix} 1 & 2 & 3&4\\ 1 & 4 & 2&3 \end{pmatrix}\begin{pmatrix} 1 & 2 & 3&4\\ 1 & 3 & 4&2 \end{pmatrix}

                             P3=\begin{pmatrix} 1 & 2 & 3&4\\ 1 & 2 & 3&4 \end{pmatrix}   =I

Hence the required number is 3.

Order=3

Example 2-: Find the order of permutation \begin{pmatrix} 1 & 4 & 2&6\\ \end{pmatrix}   .



Solution-:  Let the given permutation be P= \begin{pmatrix} 1 & 4 & 2&6\\ \end{pmatrix}

                   We can write P as P= \begin{pmatrix} 1 & 4 & 2&6\\ 4 & 2 & 6&1 \end{pmatrix}

                  P2=\begin{pmatrix} 1 & 4 & 2&6\\ 4 & 2 & 6&1 \end{pmatrix} \begin{pmatrix} 1 & 4 & 2&6\\ 4 & 2 & 6&1 \end{pmatrix}

                      =\begin{pmatrix} 1 & 4 & 2&6\\ 2 & 6 & 1&4 \end{pmatrix}

                 P3=P2.P=\begin{pmatrix} 1 & 4 & 2&6\\ 2 & 6 & 1&4 \end{pmatrix}\begin{pmatrix} 1 & 4 & 2&6\\ 4 & 2 & 6&1 \end{pmatrix}

                             =\begin{pmatrix} 1 & 4 & 2&6\\ 6 & 1 & 4&2 \end{pmatrix}

                 P4=P3.P=\begin{pmatrix} 1 & 4 & 2&6\\ 6 & 1 & 4&2 \end{pmatrix}\begin{pmatrix} 1 & 4 & 2&6\\ 4 & 2 & 6&1 \end{pmatrix}

                              =\begin{pmatrix} 1 & 4 & 2&6\\ 1 & 4& 2&6 \end{pmatrix}

P4=I (identity permutation)

Hence, order is 4.

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