Example 1: Embed a diamond in each of the 5 positions on a crown, they are distributed at the 5 equal points on a circle. If there are 3 colors of diamonds available, how many ways are there to embed them? Assume the 5 positions are indistinguishable.
Solve As shown in Figure $1-10, N_{5}=\{1,2,3,4,5\}, m=3, N_{5}$ on the group is $G=$ $\left\{P_{1}=I, P_{2}, P_{3}, P_{4}, P_{5}\right\}$, where
$P_{1}=I=(1)(2)(3)(4)(5)$, no movement,
$P_{2}=\left(\begin{array}{ll}1 & 2345)\end{array}\right.$, counterclockwise rotation $72^{\circ}$,
$P_{4}=(14253)$, counterclockwise rotation $216^{\circ}$,
$P_{5}=\left(\begin{array}{ll}15432\end{array}\right)$, counterclockwise rotation $288^{\circ}$.

By Theorem (1), the number of different embedding methods is $M=\frac{1}{5}\left(3^{5}+3+3+3+3\right)=51$.