Let M {\displaystyle {}M} be a finite set and let π {\displaystyle {}\pi } be a permutation on M {\displaystyle {}M} . Then π {\displaystyle {}\pi } is called a cycle of order r {\displaystyle {}r} , if there exists a subset Z ⊆ M {\displaystyle {}Z\subseteq M} , containing r {\displaystyle {}r} elements and such that π {\displaystyle {}\pi } is on M ∖ Z {\displaystyle {}M\setminus Z} the identity and such that π {\displaystyle {}\pi } commutes the elements of Z {\displaystyle {}Z} in a cyclic way. If Z = { z , π ( z ) , π 2 ( z ) , … , π r − 1 ( z ) } {\displaystyle {}Z=\{z,\pi (z),\,\pi ^{2}(z),\ldots ,\pi ^{r-1}(z)\}} , then we write
be a finite set and let 
be a
permutation
on . Then is called a cycle of order 
, if there exists a subset
,
containing elements and such that is on 
the identity and such that commutes the elements of 
in a cyclic way. If
,
then we write
