TITLE: Calculating the power of permutations
QUESTION [8 upvotes]: I have this permutation $A$:
$$
\left(\begin{array}{rrrrrrrrrr}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
10 & 8 & 5 & 2 & 3 & 1 & 6 & 4 & 7 & 9
\end{array}\right)
$$
I want to calculate $A^9$.  Is it ok to calculate it in this way?
$$A*A*A*A*A*A*A*A*A$$
where $A*A$ is defined as $A$ composed with $A$.  
Thanks much in advance!!!

REPLY [8 votes]: Write it as the product of disjoint cycles
$$
A = (1, 10, 9, 7, 6) (2, 8, 4) (3, 5),
$$
and then it's much easier, as a $k$-cycle has period $k$, so that
$$
A^9 = (1, 10, 9, 7, 6)^{-1} (3, 5) = (1, 6, 7 ,9 ,10) (3, 5)
=
\begin{pmatrix}
1 &2& 3& 4 &5 &6 &7 &8 &9 &10\\
6 &2& 5& 4& 3 &7& 9& 8& 10& 1\end{pmatrix}.
$$
PS Apologies, I write my permutations left-to-right.