阿贝尔群的完备化的Hopficity
Hopficity of profinite completions of abelian groups
浏览论文内容
中文总结 AI 辅助
研究阿贝尔群完备化的拓扑Hopficity问题，通过证明\(\widehat{A}\)是拓扑Hopfian当且仅当对每个素数\(p\)，\(A/pA\)有限，否定回答了Kourovka笔记本中的问题6.30。
中文摘要 AI 辅助
我们精确确定了任意阿贝尔群的完备化何时是拓扑Hopfian的。对于阿贝尔群\(A\)，我们证明\(\widehat{A}\)是拓扑Hopfian当且仅当对于每个素数\(p\)，\(A/pA\)是有限的。作为副产品，我们否定地回答了Kourovka笔记本中的问题6.30：对于两两不同的奇素数\(q_i\)，群\(\bigoplus_{i\geq1}\Z[1/q_i]\)是剩余有限且Hopfian的，但其完备化不是拓扑Hopfian的。
英文摘要
We determine exactly when the profinite completion of an arbitrary abelian group is topologically Hopfian. For an abelian group $A$, we prove that \[ \widehat A \text{ is topologically Hopfian} \quad\Longleftrightarrow\quad A/pA \text{ is finite for every prime }p. \] As a byproduct, we answer Problem 6.30 of the Kourovka Notebook in the negative: for pairwise distinct odd primes $q_i$, the group $\bigoplus_{i\geq1}\Z[1/q_i]$ is residually finite and Hopfian, whereas its profinite completion is not topologically Hopfian.
