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.
