Example 7.1 Let $A=\{1,2, \cdots, 9\}$, and let the permutation $f$ on $A$ be
$$
f=\left(\begin{array}{lllllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
3 & 4 & 7 & 6 & 9 & 2 & 1 & 8 & 5
\end{array}\right)
$$

Express $f$ as a product of disjoint cycles.
Solution: Since $f$ contains the following 4 cycles: (137), (246), (59), (8), we have
$$
f=(137)(246)(59)(8) \text {. }
$$

Theorem 7.11 Let $A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be an $n$-element set, and let $S_{A}$ denote the set of all permutations on $A$. Define a binary operation “” on $S_{A}$ as follows: for any $f, g \in S_{A}, f \circ g$ is a permutation on $A$ such that
$$
f \circ g\left(a_{k}\right)=f\left(g\left(a_{k}\right)\right) \quad(k=1,2, \cdots, n),
$$

then the algebraic system $\left(S_{A},{ }^{\circ}\right)$ is a group.