$a_1,a_1a_2,\cdots,a_1a_2\ldots a_p$构成模$p$的完全剩余系

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设 $p$ 是素数, 证明: 存在正整数 $1,2,\dots,p$ 的一个排列 $a_1,a_2,\dots,a_p$ 使得 $a_1,a_1a_2,\cdots,a_1a_2\ldots a_p$ 构成模 $p$ 的完全剩余系.
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