2008 Putnam A6

Prove that there exists a constant $c>0$ such that in every

nontrivial finite group $G$ there exists a sequence of length

at most $c \log |G|$ with the property that each element of $G$

equals the product of some subsequence. (The elements of $G$ in the

sequence are not required to be distinct. A *subsequence*

of a sequence is obtained by selecting some of the terms,

not necessarily consecutive, without reordering them; for

example, $4, 4, 2$ is a subsequence of $2, 4, 6, 4, 2$, but

$2, 2, 4$ is not.)

证明存在常数 $c>0$ 使得在每个

非平凡有限群 $G$ 存在一个长度序列

至多 $c \log |G|$ 且 $G$ 的每个元素

等于某个子序列的乘积。 ($G$ 中的元素

序列不要求是不同的。 A *子序列*

序列的 是通过选择一些项来获得的，

不一定是连续的，无需重新排序；为了

例如，$4, 4, 2$ 是 $2, 4, 6, 4, 2$ 的子序列，但是

$2, 2, 4$ 不是。)