On Twisted Subgroups and Bol Loops of Odd Order

关于在可换环上扭曲Chevalley群的正规子群

On Normal Subgroups of Twisted Chevalley Groups over Commutative Rings

本文研究了扭曲Chevalley群在可换环上的结构，证明了初等同余子群的正规性及子群分类，同时探讨了半局部环下的群分解性质。

A class of Bol loops with a subgroup of index two

Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto (x^iy^j)^k$, where $i$, $j$, $k\in\{-1, 1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times\{i\})\times(G\times\{j\})$, for $i$, $j\in C_2$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovský: On the uniqueness of loops $M(G,2)$.

关于扭曲右角Artin群的子群可分性与成员判定问题

On Subgroup Separability and Membership Problems in Twisted Right-Angled Artin Groups

本文仅通过定义混合图刻画了扭曲右角Artin群的子群可分性，推广了经典结果，并证明了凝聚群的子群成员判定问题可判定，同时解决了特定锥图族对应的有理与子半群成员判定问题。

Twisted Poincare Series and Zeta functions on finite quotients of buildings

In the case where $G=$SL$_{2}(F)$ for a non-archimedean local field $F$ and $Γ$ is a discrete torsion-free cocompact subgroup of $G$, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of $G$ by the action of $Γ$, and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of $G$. We show how this can be generalised to other split simple algebraic groups of rank two over $F$, and formulate a conjecture about how this might be generalised to groups of higher rank.

一种研究扭曲同调稳定性的新方法，及其在同余子群上的应用

A new approach to twisted homological stability, with applications to congruence subgroups

本文提出证明扭曲同调稳定性的新方法，用于对称群与一般线性群，并将 Borel 关于有限指数子群有理上同调不变的定理推广到多环 GL_n 的扭曲系数情形。

扭曲虚拟辫子群

Twisted Virtual Braid Group

本文研究扭曲虚拟辫子群TVBₙ的子群结构与半直积分解，确定了TVPₙ、TVHₙ等子群的生成元、定义关系及分解形式，得出VPₙ是TVPₙ子群的结论，丰富了虚拟辫子群的代数结构研究。

扭Chevalley群的某些性质

Some Properties of Twisted Chevalley Groups

本论文研究交换环上扭Chevalley群的结构性质，探讨了相对初等子群的正规性、被初等子群正规化的子群分类，以及在环扩张中正规化子的重合性，证明了伴随型群的正规化子恰好等于原群。

$\boldsymbol{\mathrm{Sym}(p^n)}$ 的 Sylow $\boldsymbol{p}$-子群及其相伴李代数中的正规性条件

Normality conditions in the Sylow $\boldsymbol{p}$-subgroup of $\boldsymbol{\mathrm{Sym}(p^n)}$ and its associated Lie algebra

本文研究了对称群 $\mathrm{Sym}(p^n)$ 的 Sylow $p$-子群 $W_n$ 的正规子群结构，明确确定了其上、下中心序列，并在相伴李代数中给出对应描述，从而重新证明了两种语境下中心序列项相等，还计算了从初等阿贝尔正则子群出发的正规化子链的增长。

Twisted Automorphisms of Right Loops

In this paper we make an attempt to study right loops $(S, o)$ in which, for each $y\in S$, the map $σ_y$ from the inner mapping group $G_S$ of $(S, o)$ to itself given by $σ_y (h)(x) o\ h(y)= h(xoy)$, $x\in S, h\in G_S$ is a homomorphism. The concept of twisted automorphisms of a right loop and also the concept of twisted right gyrogroup appears naturally and it turns out that the study is almost equivalent to the study of twisted automorphisms and a twisted right gyrogroup. A representation theorem for twisted right gyrogroup is established. We also study relationship between twisted gyrotra…

Congruence subgroups and twisted cohomology of SL_n(F[t])

Let F be a field of characteristic zero and let V be an irreducible representation of SL_n(F). In this paper, we compute the first cohomology of SL_n(F[t]) with coefficients in V. It agrees with H^1(SL_n(F),V) if V is not the adjoint representation, while if V = Ad, the two groups differ by an F-vector space X. We show that if n=2, X is infinite dimensional, while if n>2, dim X = 1. We also study the abelianization of the kernel of the map SL_n(F[t])-->SL_n(F) given by setting t=0, where now F is any field. We conjecture that this abelianization is the adjoint representation sl_n(F) if n>2 and…

Holomorphic Structure of Middle Bol Loops

A loop $(Q,\cdot,\backslash,/)$ is called a middle Bol loop if it obeys the identity $x(yz\backslash x)=(x/z)(y\backslash x)$. To every right (left) Bol loop corresponds a middle Bol loop via an isostrophism. In this paper, the structure of the holomorph of a middle Bol loop is explored. For some special types of automorphisms, the holomorph of a commutative loop is shown to be a commutative middle Bol loop if and only if the loop is a middle Bol loop and its automorphism group is abelian and a subgroup of both the group of middle regular mappings and the right multiplication group. It was fou…

扭曲直角Artin群的Droms定理

Droms Theorems for twisted right-angled Artin groups

本文刻画了有限生成子群仍为扭曲直角Artin群的扭曲直角Artin群，基于定义图分类了该类群的凝聚性，并解决了其一个重要子类的同构问题。

Palindromes in finite groups and the Explorer-Director game

In this paper, we use the notion of twisted subgroups (i.e., subsets of group elements closed under the binary operation $(a,b) \mapsto aba$) to provide the first structural characterization of optimal play in the Explorer-Director game, introduced as the Magnus-Derek game by Nedev and Muthukrishnan and generalized to finite groups by Gerbner. In particular, we reduce the game to the problem of finding the largest proper twisted subgroup, and as a corollary we resolve the Explorer-Director game completely for all nilpotent groups.

On reduced twisted group C*-algebras that are simple and/or have a unique trace

We study the problem of determining when the reduced twisted group C*-algebra associated with a discrete group G is simple and/or has a unique tracial state, and present new sufficient conditions for this to hold. One of our main tools is a combinatorial property, that we call the relative Kleppner condition, which ensures that a quotient group G/H acts by freely acting automorphisms on the twisted group von Neumann algebra associated to a normal subgroup H. We apply our results to different types of groups, e.g. wreath products and Baumslag-Solitar groups.