Counting conjugacy classes of subgroups in a finitely generated group

计算例外李群中有限阶元素的共轭类数

Counting conjugacy classes of elements of finite order in exceptional Lie groups

本文针对连通且单连通的例外李群，提出计算有限阶元素共轭类数及其按不同特征值分类数量的系统算法，并给出 $N(G,m)$、$N(G_2,m,s)$ 和 $N(F_4,m,s)$ 的显式结果。

Counting Conjugacy Classes of Elements of Finite Order in Lie Groups

Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define $N(G,m)$ to be the number of conjugacy classes of elements of finite order $m$ in a Lie group $G$, and $N(G,m,s)$ to be the number of such classes whose elements have $s$ distinct eigenvalues or conjugate pairs of eigenvalues. What is $N(G,m)$ for $G$ a unitary, orthogonal, or symplectic group? What is $N(G,m,s)$ for these groups? For some cases, the first question was answered a few decades ago via group-theoretic techniques. It appears that the second question has not been asked before; here it …

Solvable conjugacy class graph of groups

In this paper we introduce the graph $Γ_{sc}(G)$ associated with a group $G$, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of $G$ and two distinct conjugacy classes $C, D$ are adjacent if there exist $x \in C$ and $y \in D$ such that $\langle x, y\rangle$ is solvable. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number~$d$, and we find explicitly the list of s…

近阿贝尔群中的共轭类增长

Conjugacy Class Growth in Virtually Abelian Groups

本文研究有限生成近阿贝尔群中固定共轭类在Cayley球内的计数增长，证明其渐近等价于多项式，并揭示仿射Coxeter群中增长度与反射长度的等价关系。

Hecke群中倒数双曲元素的计数

Counting Reciprocal Hyperbolic Elements in Hecke Groups

本文研究Hecke群中倒数双曲元素的计数，确定了其素共轭类数量的渐近增长率与极限常数。

结合Burnside过程与重要性采样计数群轨道数量

Counting the number of group orbits by marrying the Burnside process with importance sampling

本研究提出结合Burnside过程与重要性采样的通用群轨道近似计数新算法，应用于单位上三角群时可高效估算其共轭类数量。