论文标题
有限正交和符号组的力量：生成函数方法
Powers in finite orthogonal and symplectic groups: A generating function approach
论文作者
论文摘要
对于整数$ m \ geq 2 $和有限的组$ g $，如果满足$ a^m =α$，则为$ m $ the的元素$α\在g $中的$ m $ th power。在本文中，我们将在订单$ q $的字段上$ g $是有限的符号或正交组时处理的情况。我们介绍了$ m^*$ - 功率SRIM多项式的概念。与$ m $ - power多项式的概念合并，我们提供了常规半神经，半imple，semimple，cyclic和pradion元素的共轭类别的完整分类，即$ m $ $ th powers，当$ m $ th powers，当$（m，q）= 1 $时。此处的方法是产生功能，如杰森·富尔曼（Jason Fulman），彼得·M·诺伊曼（Peter M.作为副产品，我们从生成功能方面获得了相应的概率。
For an integer $M\geq 2$ and a finite group $G$, an element $α\in G$ is called an $M$-th power if it satisfies $A^M=α$ for some $A\in G$. In this article, we will deal with the case when $G$ is finite symplectic or orthogonal group over a field of order $q$. We introduce the notion of $M^*$-power SRIM polynomials. This, amalgamated with the concept of $M$-power polynomial, we provide the complete classification of the conjugacy classes of regular semisimple, semisimple, cyclic and regular elements in $G$, which are $M$-th powers, when $(M,q)=1$. The approach here is of generating functions, as worked on by Jason Fulman, Peter M. Neumann, and Cheryl Praeger in the memoir "A generating function approach to the enumeration of matrices in classical groups over finite fields". As a byproduct, we obtain the corresponding probabilities, in terms of generating functions.
