2016 Putnam A5

Suppose that $G$ is a finite group generated by the two elements $g$ and $h$, where the order of $g$ is odd. Show that every element of $G$ can be written in the form
$$g^{m_1} h^{n_1} g^{m_2} h^{n_2} \cdots g^{m_r} h^{n_r}$$
with $1 \leq r \leq |G|$ and $m_1, n_1, m_2, n_2, \ldots, m_r, n_r \in \{-1, 1\}$.
(Here $|G|$ is the number of elements of $G$.)

\,

假设$G$是由$g$和$h$两个元素生成的有限群，其中$g$的阶数为奇数。表明 $G$ 的每个元素都可以写成以下形式

$$

g^{m_1} h^{n_1} g^{m_2} h^{n_2} \cdots g^{m_r} h^{n_r}

$$

与 $1 \leq r \leq |G|$ 和 $m_1, n_1, m_2, n_2, \ldots, m_r, n_r \in \{-1, 1\}$。

(这里$|G|$是$G$的元素数量。)

\,