A group $G$ is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups $G$ that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to $G$. We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: first, we show any proper self-embedding of $G$ with finite-index characteristic image arises by pulling back an endomorphism of the abelianization; secondly, we prove special cases (for normal subgroups) of conjectures of Benjamini and Nekrashevych–Pete regarding the classification of scale-invariant groups.

中文翻译：

---

正常和有限非 co-Hopfian 群的结构

如果群 $G$ 不包含任何与自身同构的适当（有限索引）子群，则它是（有限）co-Hopfian。我们研究了有限生成群$G$，它允许一个适当的正规有限指数子群的下降链，每个子群与$G$同构。我们证明，直到有限索引，这些总是通过从自由阿贝尔商中拉回子群链来获得的。我们给出了两个应用：首先，我们展示了通过拉回阿贝尔化的内同态而产生的具有有限索引特征图像的 $G$ 的任何适当的自嵌入；其次，我们证明了 Benjamini 和 Nekrashevych-Pete 关于尺度不变组分类的猜想的特殊情况（对于正常子组）。