AI 中文总结
针对《库罗夫卡笔记本》中关于有限群分解的问题，G. M. 伯格曼给出\(k = 3\)反例并提问，本文给出\(k \geq 3\)的反例，解决了部分问题，\(k = 2\)时问题仍待解决。
AI 中文摘要
在《库罗夫卡笔记本》的问题19.35中，M. H. 胡什曼德提出，对于有限群\(G\)及分解\(|G| = n_1 \cdots n_k\)，是否总能找到\(G\)的子集\(A_1, \dots, A_k\)满足\(|A_i| = n_i\)且\(G = A_1 \cdots A_k\)。G. M. 伯格曼给出\(k = 3\)的反例并询问更大\(k\)时的情况。本文给出\(k \geq 3\)的反例，\(k = 2\)时问题仍未解决。
英文摘要
In this note we answer some questions of G. M. Bergman concerning factorizations of finite groups. Such questions originated when M. H. Hooshmand asked whether, given a finite group $G$ and a factorization $|G| = n_1 \cdots n_k$, one can always find subsets $A_1, \dots , A_k$ of $G$ with $|A_i| = n_i$ such that $G = A_1 \cdots A_k$. This was Question 19.35 of the Kourovka Notebook. G. M. Bergman provided a counterexample for $k=3$, and Kabenyuk provided counterexamples for all $k \geq 3$. We present these counterexamples with direct proofs that mirror Bergman's original $k=3$ argument, and we answer another recent question of Bergman.
