Let $I=[0,1], G=\{(x, y) \mid x \in I$, $y \in I\}$, find all mappings $f$ from $G$ to $I$ such that for any $x, y, z \in I$, the following conditions hold:
(1) $f[f(x, y), z]=f(x, f(y, z)]$,
(2) $f(x, 1)=x, f(1, y)=y$;
(3) $f(z x, z y)=z^{\mathrm{k}} f(x, y)$, where $k$ is a positive number independent of $x, y, z$.
Solve: From (3) and (2) we have
$$
\begin{array}{l}
f(x, y)=y^{k} f\left(\frac{x}{y}, 1\right)=y^{k-1} x, \\
x \leqslant y_{3} \\
f(x, y)=x^{k} f\left(1, \frac{y}{x}\right)=x^{k-1} y, \\
x \geqslant y .
\end{array}
$$

Take $0 < x < y$.
\end{array}\right. \\
f_{2}(x, y)=x y^{\prime} \text{.} \\
\end{array}
$$

It is easy to verify that both functions satisfy conditions (1)-(3).