On perfect order subsets in finite groups

关于Baumslag-Solitar群的子群空间I：完美核与表型

On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype

本文从拓扑和动力学角度研究Baumslag-Solitar群的子群空间，确定其完美核，并引入由算术函数定义的“表型”划分，证明共轭作用在各部分上的拓扑传递性，并识别最大紧致不变子空间。

On a perfect isometry between principal $p$-blocks of finite groups with cyclic $p$-hyperfocal subgroups

Let $G$ be a finite group with a Sylow $p$-subgroup $P$. We prove that the principal $p$-blocks of $G$ and $N_G(P)$ are perfectly isometric under the assumption $G$ has a cyclic $p$-hyperfocal subgroup.

对称群$S_n$在Cayley和图中的子群完美码

Subgroup perfect codes of $S_n$ in Cayley sum graphs

该研究证明对称群$S_n$不存在真子群完美码，并以此为基础给出了交错群$A_n$子群完美码的完整刻画，推进了Cayley和图中群论与编码理论交叉领域的研究。

Perfect models for finite Coxeter groups

A model for a finite group is a set of linear characters of subgroups that can be induced to obtain every irreducible character exactly once. A perfect model for a finite Coxeter group is a model in which the relevant subgroups are the quasiparabolic centralizers of perfect involutions. In prior work, we showed that perfect models give rise to interesting examples of $W$-graphs. Here, we classify which finite Coxeter groups have perfect models. Specifically, we prove that the irreducible finite Coxeter groups with perfect models are those of types $\mathsf{A}_{n}$, $\mathsf{B}_n$, $\mathsf{D}_…

Hamiltonian Groups with Perfect Order Classes

A finite group is said to have "perfect order classes" if the number of elements of any given order is either zero or a divisor of the order of the group. The purpose of this note is to describe explicitly the finite Hamiltonian groups with perfect order classes. We show that a finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic to the direct product of the quaternion group of order $8$, a non-trivial cyclic $3$-group and a group of order at most $2$. Theorem. A finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic either to $Q\ti…

On subgroup perfect codes in Cayley sum graphs

A perfect code $C$ in a graph $Γ$ is an independent set of vertices of $Γ$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $Γ$ is a set of vertices of $Γ$ such that every vertex of $Γ$ is adjacent to a unique vertex in $C$. Let $G$ be a finite group and $X$ a normal subset of $G$. The Cayley sum graph $\mathrm{CS}(G,X)$ of $G$ with the connection set $X$ is the graph with vertex set $G$ and two vertices $g$ and $h$ being adjacent if and only if $gh\in X$ and $g\neq h$. In this paper, we give some necessary conditions of a subgroup of…