关于置换特征标的独立性
On the independence of permutation characters
AI总结：
该研究证明有限群G的子群共轭类代表元对应的置换特征标线性无关当且仅当G是循环群，且有限非可解群不满足Kourovka Notebook第11.9题的要求。
AI中文摘要：
设G为有限群，对每个子群H≤G，记π_H为G在H的左陪集上作用的置换特征标。我们证明：当且仅当G是循环群时，由G的子群共轭类代表元H对应的特征标π_H构成的集合线性无关。特别地，不存在有限非可解群具备Kourovka Notebook第11.9题所要求的性质。该证明仅用到陪集作用的不动点公式与初等三角矩阵论证。
英文摘要：
Let $G$ be a finite group. For every subgroup $H\leq G$, let $π_H$ be the permutation character of the action of $G$ on the left cosets of $H$. We prove that the characters $π_H$, with $H$ running through representatives of the conjugacy classes of subgroups of $G$, are linearly independent if and only if $G$ is cyclic. In particular, no finite insoluble group has the property asked for in Kourovka Notebook Problem~11.9. The proof uses only the fixed-point formula for a coset action and an elementary triangular-matrix argument.
