关于自同构对不可忽略部分元素求逆或平方的有限群的一个注记
A note on finite groups with an automorphism inverting or squaring a non-negligible fraction of elements
摘要：
我们证明了对于有限群$G$，$G$的交换概率可以通过一个在元素的最大分数中的函数从下面以非平凡的方式显式地有界.由$G$的自同构平方。使用这些界限连同Guralnick和Robinson的结果给出了上界指数的拟合子群的$G$下的两个条件下，$G$有一个自同构反演resp。在$G$中平方至少$ρ|G|$多个元素，用于固定$ρ\in\left(0,1\right]$。这是对作者以前结果的改进。
英文摘要：
We show that for a finite group $G$, the commuting probability of $G$ can be explicitly bounded from below in a nontrivial way by a function in the maximum fraction of elements inverted resp. squared by an automorphism of $G$. Using these bounds together with a result of Guralnick and Robinson gives upper bounds on the index of the Fitting subgroup of $G$ under each of the two conditions that $G$ have an automorphism inverting resp. squaring at least $ρ|G|$ many elements in $G$, for $ρ\in\left(0,1\right]$ fixed. This is an improvement on previous results of the author.
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