The following problem was originally posed by B.H. Neumann and H. Neumann. Suppose that a group $G$ can be generated by $n$ elements and that $H$ is a homomorphic image of $G$. Does there exist, for every generating $n$-tuple $(h_1,\ldots, h_n)$ of $H$, a homomorphism $\vartheta \colon G \to H$ and a generating $n$-tuple $(g_1,\ldots,g_n)$ of $G$ such that $(g_1^\vartheta,\ldots,g_n^\vartheta) = (h_1,\ldots,h_n)$? M.J. Dunwoody gave a negative answer to this question, by means of a carefully engineered construction of an explicit pair of soluble groups. Via a new approach we produce, for $n = 2$, infinitely many pairs of groups $(G,H)$ that are negative examples to the Neumanns' problem. These new examples are easily described: $G$ is a free product of two suitable finite cyclic groups, such as $C_2 \ast C_3$, and $H$ is a suitable finite projective special linear group, such as $\mathrm{PSL}(2,p)$ for a prime $p \ge 5$. A small modification yields the first negative examples $(G,H)$ with $H$ infinite.

中文翻译：

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生成无法提升的投影特殊线性群对

以下问题最初是由 BH Neumann 和 H. Neumann 提出的。假设一组$G$ 可以由$n$ 个元素生成，$H$ 是$G$ 的同态图像。是否存在，对于每个生成 $n$-元组 $(h_1,\ldots, h_n)$ 的 $H$，同态 $\vartheta \colon G \to H$ 和生成 $n$-tuple $(g_1 ,\ldots,g_n)$ 的 $G$ 使得 $(g_1^\vartheta,\ldots,g_n^\vartheta) = (h_1,\ldots,h_n)$？MJ Dunwoody 通过精心设计的一对明确的可溶基团对这个问题给出了否定的答案。通过一种新方法，对于 $n = 2$，我们产生无限多对组 $(G,H)$，它们是 Neumanns 问题的反例。这些新例子很容易描述：$G$ 是两个合适的有限循环群的自由乘积，例如 $C_2 \ast C_3$，而 $H$ 是一个合适的有限射影特殊线性群，例如 $\mathrm{PSL}(2,p)$ 表示素数 $p \ge 5$。一个小的修改产生第一个负样本 $(G,H)$，$H$ 无限。