Inverse of any Permutation \(P\) is a another Permutation \(P^{-1}\) such that Applying \(P^{-1}\) after \(P\) or Applying \(P\) after \(P^{-1}\) gives back the Identity Permutation \(I\). That is
\(PP^{-1}=P^{-1}P=I\)
Every Permutation has a Unique Inverse Permutation.
The Elements of Permutation Cycle(s) that represent the Permutation \(P^{-1}\) are in Reverse Order of the Elements of Permutation Cycle(s) that represent the Permutation \(P\). That is
if the Permutation \(P\) is given by Permutation Cycles \((5\hspace{.1cm}3\hspace{.1cm}2\hspace{.1cm}6)\) \((1\hspace{.1cm}4\hspace{.1cm}9)\)
then the Permutation Cycles of the Permutation \(P^{-1}\) is given by \((6\hspace{.1cm}2\hspace{.1cm}3\hspace{.1cm}5)\) \((9\hspace{.1cm}4\hspace{.1cm}1)\).
Order of any Permutation \(P\) is One more than the Number of times the Permutation \(P\) has to be Applied / Multiplied to itself to get back the Identity Permutation \(I\). That is
if \(P^N=I\) then \(N\) is the Order of Permutation \(P\).