AI 中文总结
本文解决Kourovka问题21.88，证明无奇数阶有限群的交换概率为1/17，还结合Burnside同余排除更小奇素数的对应情况，进而研究p=97的未解决情形。
AI 中文摘要
我们对Kourovka Notebook问题21.88给出否定答案：不存在阶为奇数的有限群，其交换概率cp(G)为1/17。这一结论源于一个结构定理：对任意奇素数p，若cp(G)=1/p，则G的Sylow p-子群是正规阿贝尔群。结合奇阶群共轭类数的Burnside同余式，该结论还排除了所有小于97的奇素数p对应的cp(G)=1/p的情况。我们进一步研究该同余式未排除的下一个未解决情形，即p=97。
英文摘要
We give a negative answer to Kourovka Notebook Problem 21.88: no finite group of odd order has commuting probability $1/17$. This follows from a structural theorem asserting that, whenever $p$ is an odd prime and $cp(G)=1/p$, a Sylow $p$-subgroup of $G$ is normal and abelian. Together with Burnside's congruence for the number of conjugacy classes of a group of odd order, this also excludes $cp(G)=1/p$ for every odd prime $p<97$. We further study the next unresolved case not excluded by this congruence, namely $p=97$.
