强制群与Frattini覆盖
Forcible groups and Frattini covers
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中文总结 AI 辅助
本文补充Thompson-Mann经典结果并回答相关问题，证明有限群可强制的充要条件，同时给出阿贝尔p-群在两类p-群下的相对可强制性结果。
中文摘要 AI 辅助
若有限群G的每个有限覆盖都包含一个同构于有限群H的子群，则称G*强制*有限群H；若存在某个有限群G强制H，则称H是*可强制的*。本文补充了Thompson-Mann的经典结果，并回答了第三作者近期提出的问题：证明有限群可强制当且仅当它是阿贝尔群且其Sylow子群是初等循环群。同时，本文还针对阿贝尔p-群，在幂p-群和有界幂零类p-群的框架下证明了相对可强制性结果。
英文摘要
We say that a finite group $G$ *forces* a finite group $H$ if every finite cover of $G$ contains a subgroup isomorphic to $H$, and we say $H$ is *forcible* if some finite group $G$ forces $H$. Complementing a classical result of Thompson--Mann, and answering a recent question of the third author, we show that a finite group is forcible if and only if it is abelian and its Sylow subgroups are elementary-by-cyclic. We also prove relative forcibility results for abelian $p$-groups in the settings of powerful $p$-groups and $p$-groups of bounded nilpotency class.
