AI 中文总结
本文研究有限单射影特殊线性群的全3-闭包性质，明确了PSL₂(q)、PSL₃(q)、PSL₄(q)及n≥5且q>2时PSLₙ(q)的判定条件，还肯定回答了《Kourovka Notebook》中的问题20.2。
AI 中文摘要
有限群是全3-闭包的，当且仅当它的每个忠实置换表示都是3-闭包的。我们针对有限单射影特殊线性群研究该性质，证明：当且仅当q≥7为素数时，PSL₂(q)是全3-闭包的；当且仅当q=3，或q为素数且q≡2 mod 3时，PSL₃(q)是全3-闭包的；进一步证明PSL₄(q)永远不是全3-闭包的，且当n≥5、q>2时，PSLₙ(q)也不是全3-闭包的。在PSLₙ(q)族中，仅n≥5的PSLₙ(2)仍未解决，该研究还肯定回答了《Kourovka Notebook》中的问题20.2。
英文摘要
A finite group is totally $3$-closed if every faithful permutation representation of it is $3$-closed. We study this property for the finite simple projective special linear groups. We prove that $\PSL_2(q)$ is totally $3$-closed if and only if $q\geq 7$ is prime, and that $\PSL_3(q)$ is totally $3$-closed if and only if either $q=3$, or $q$ is prime and $q\equiv 2\pmod 3$. We further prove that $\PSL_4(q)$ is never totally $3$-closed and that $\PSL_n(q)$ is not totally $3$-closed whenever $n\geq 5$ and $q>2$. Within the family $\PSL_n(q)$, only the groups $\PSL_n(2)$ with $n\geq 5$ remain unresolved. In particular, this answers Problem~20.2 of the Kourovka Notebook affirmatively.
Comments26 pages. Comments welcome!
