对称群与交错群中错位排列的共轭类
On Conjugacy Classes of Derangements in Symmetric and Alternating Groups
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中文总结 AI 辅助
本文证明了对称群与交错群中错位排列共轭类的两个猜想，还肯定回答了伯特伦的特定问题，并得出错位排列共轭类生成对称或交错群的相关结论。
中文摘要 AI 辅助
本文证明了伯内斯（Timothy Burness）与富萨里（Marco Fusari）提出的两个关于对称群与交错群中错位排列的共轭类的幂和乘积的猜想：（1）证明了存在对称群$S_n$的两个错位排列共轭类$C$和$D$，使得$S_n = C^2 \cup CD$；（2）证明了当$n \equiv 3 \pmod{4}$时，交错群$A_n$存在一个错位排列共轭类$C$，使得$C^2 = A_n$。事实上，关于第二个猜想的结果适用范围更广，还在特定情形下肯定回答了伯特伦（Edward Bertram）提出的一个问题。此外，证明了对称群$S_n$（或交错群$A_n$）的任意错位排列共轭类$C$，除非是无不动点对换的共轭类，都包含一对生成$S_n$（或$A_n$，当$C$属于$A_n$时）的元素。
英文摘要
In this article, we prove two conjectures of Burness and Fusari [Timothy Burness and Marco Fusari, On derangements in simple permutation groups, Forum Math. Sigma 13 (2025)] concerning the powers and products of conjugacy classes of derangements in the symmetric and alternating groups: (1) We show that there exist two conjugacy classes $C$ and $D$ of derangements in $S_n$ such that $S_n=C^2\cup CD$, and (2) We show that there exists a conjugacy class $C$ of derangements in $A_n$ such that $C^2=A_n$, whenever $n\equiv 3\;(\text{mod}\;4)$. In fact, our result concerning the second conjecture holds in a considerably more general setting, which also answers affirmatively a question posed by Bertram [Edward Bertram, Even permutations as a product of two conjugate cycles, J. Comb. Theory, Ser. A 12 (1972), 368-380] in a particular case. Moreover, we show that any conjugacy class $C$ of derangements in $S_n$ (resp. $A_n$) contains a pair of elements that generate $S_n$ or $A_n$ (resp. $A_n$), unless $C$ is the conjugacy class of fixed-point-free involutions.
