有限对称群中素数平方阶元素的比例
Proportion of Elements of Prime Square Order in a Finite Symmetric Group
摘要: 为了构造G的生成集，需要寻找特殊类型的元素，这些元素通常是随机寻找。此外，为了了解寻找的复杂性，需要估计各种元素的比例。令n是正整数，设有限集合Ω={1,2,⋯,n}，记Sym(Ω)是Ω上全体置换所组成的群，称作Ω上的对称群。令p是素数，设Ρn(p2)是对称群Sym(Ω)中所有p2阶元素所组成的集合，ρn(p2)是对称群Sym(Ω)中p2阶元素所占比例，本文首先对p2阶元素在对称群Sym(Ω)中所能表现的形式进行分析，找到其所有的表现形式，接着找出相应形式下p2元素比例的递归公式；在此基础上，通过合理构造对称群Sym(Ω)中元素数，并结合归纳法，最终得出ρn(p2)的上界表达式。
Abstract: In order to construct a generating set for G, we need to find certain kinds of elements, which are usually found randomly. In addition, in order to evaluate the complexity of the finding process, we need to estimate the proportion of elements of different kinds. Let n be a positive integer, and the finite setΩ={1,2,⋯,n}. LetSym(Ω)be the group of all permutations on Ω, called the symmetry group on Ω. Let p be a prime andPn(p2)be the set of all elements of order p2 in the symmetry groupSym(Ω),ρn(p2)is the proportion of elements of order p2 in the symmetry groupSym(Ω). In this paper, first analyze the forms of elements of order p2 in the symmetric groupSym(Ω), and find all the forms; then need to find the recursive formula of the proportion of elements of order p2 in all the forms; based on reasonable construction of the number of elements in the symmetry groupSym(Ω)and combined with induction, obtain the upper bound expression ofρn(p2).
文章引用：陈守双, 钟丽萍, 张师贤. 有限对称群中素数平方阶元素的比例[J]. 理论数学, 2024, 14(6): 373-386. https://doi.org/10.12677/pm.2024.146257

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