Diego Rattaggi
Let $G_1 \times G_2$ be a subgroup of $\mathrm{SO}_3(\mathbb{R})$ such that the two factors $G_1$ and $G_2$ are non-trivial groups. We show that if $G_1 \times G_2$ is not abelian, then one factor is the (abelian) group of order 2, and the other factor is non-abelian and contains an element of order 2. There exist finite and infinite such non-abelian subgroups.
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For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of $\mathrm{G}_2(p^n)$ or $\mathrm{PSU}_4(p^n)$ with $n\in\mathbb{N}$, we determine all saturated fusion systems supported on $S$ up to isomorphism.
We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for $\mathrm{Out}(F_n)$.
Recognition and constructive membership for discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$
本文提出判定SL₂(ℝ)有限生成子群离散性、解决离散子群构造性成员资格问题并计算Fuchsian群基本域的算法，且已在Magma中实现。
On finite presentability of some partial Torelli subgroups of Aut(F_n)
本文研究了某些部分Torelli子群在Aut(F_n)中的有限可现性问题，证明了包含IA_n的特定子群的有限可现性。
Class numbers and nilpotent subgroups of $\mathrm{PGL}(2,q)$
本文证明对于某些奇素数幂 $q$，$\mathrm{PGL}(2,q)$ 的共轭类数大于其最大幂零子群的阶，从而否定了 Liebeck 和 Pyber 的一个问题。
For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exotic.
Fixed subgroups in Artin groups
本文研究大型Artin群自同构的固定子群，定义了自然子群并证明固定子群为有限生成Artin群，秩有界，并给出该子群的几何刻画。
Let $\frak{F}$ be a class of finite groups. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{F_{\mathrm s}}$-quasinormal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-permutable in $G$ and $(H\cap T)H_G/H_G$ is contained in the $\frak{F}$-hypercenter $Z_\infty^\mathfrak{F}(G/H_G)$ of $G/H_G$. In this paper, we investigate further the influence of $\mathfrak{F_{\mathrm s}}$-quasinormality of some subgroups on the structure of finite groups. New characterization of some classes of finite groups are obtained.
We study invariant random subgroups (IRSs) of semidirect products $G = A \rtimes Γ$. In particular, we characterize all IRSs of parabolic subgroups of $\mathrm{SL}_d(\mathbb{R})$, and show that all ergodic IRSs of $\mathbb{R}^d \rtimes \mathrm{SL}_d(\mathbb{R})$ are either of the form $\mathbb{R}^d \rtimes K$ for some IRS of $\mathrm{SL}_d(\mathbb{R})$, or are induced from IRSs of $Λ\rtimes \mathrm{SL}(Λ)$, where $Λ< \mathbb{R}^d$ is a lattice.
We will give a general criterion - the existence of an $F$-obstruction - for showing that a subgroup of $\mathrm{PL}_+ I$ does not embed into Thompson's group $F$. An immediate consequence is that Cleary's "golden ratio" group $F_τ$ does not embed into $F$. Our results also yield a new proof that Stein's groups $F_{p,q}$ do not embed into $F$, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of $F$-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the …
On intersections of locally quasiconvex subgroups
该研究针对双曲群的无挠局部拟凸子群，证明了其交的秩可由原二子群的秩界定，补充了双曲群子群交的秩相关结论。
We study embeddings of $\mathrm{PSL}_2(p^a)$ into exceptional groups $G(p^b)$ for $G=F_4,E_6,{}^2\!E_6,E_7$, and $p$ a prime with $a,b$ positive integers. With a few possible exceptions, we prove that any almost simple group with socle $\mathrm{PSL}_2(p^a)$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$ and $E_7$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type $A_1$ inside the algebraic group. Together with a recent result of Burness and Testerman for $p$ the Coxeter number plus one, this proves t…
Let $G$ be a simple algebraic group over an algebraically closed field $k$, where $\mathrm{char}\, k$ is either 0 or a good prime for $G$. We consider the modality $\mathrm{mod}(B : \mathfrak u)$ of the action of a Borel subgroup $B$ of $G$ on the Lie algebra $\mathfrak u$ of the unipotent radical of $B$, and report on computer calculations used to show that $\mathrm{mod}(B:\mathfrak u) = 20$, when $G$ is of type $\mathrm E_8$. This completes the determination of the values for $\mathrm{mod}(B:\mathfrak u)$ for $G$ of exceptional type.
Principal congruence subgroups in the infinite rank case
本研究将1960年代关于整数一般线性群与特殊线性群的经典结果推广到无限生成自由阿贝尔群的自同构群，给出其正规生成元、极大正规子群分类、主同余子群正规生成元及Brenner梯关系的类比结论。
The maximal subgroups of the Monster
本文利用Python包mmgroup的可复现计算，解决了魔群极大子群分类的遗留问题，发现两个新极大子群，纠正了此前的错误论断，完成了全部散在单群的极大子群分类。
On $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups of finite groups and related formations
本文定义了有限群中的${\rm K}$-$\bb P_{t}$-次正规子群概念，研究了这类子群的性质，并刻画了所有Sylow子群均为${\rm K}$-$\bb P_{t}$-次正规的有限群类的相关性质，丰富了有限群子群结构与群系理论的研究内容。
Subgroups of symmetric groups: enumeration and asymptotic properties
本文证明对称群 $\mathrm{S}_n$ 有 $2^{n^2/16+o(n^2)}$ 个子群，解决了 Pyber 1993 年的猜想，并推导了各类子群（含 $p$-子群）数量的渐近紧致上下界及随机子群的性质。
Fusion Systems on Sylow $3$-subgroups of Fischer and Monster sporadic groups -- I
本文研究菲舍尔和莫诺斯特奇奇异性群中3-子群的无核融合系统分类，证明3-群不支持奇异融合系统，为后续奇素数p的奇异性群Sylow子群融合系统分类奠定基础。
Let ${\mathfrak{X}}$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Denote by ${\mathrm{k}}_{\mathfrak{X}}(G)$ the number of conjugacy classes ${\mathfrak{X}}$-maximal subgroups of a finite group $G$. The natural problem to describe up to conjugacy ${\mathfrak{X}}$-maximal subgroups of a given finite group is complicated by the fact that it is not inductive. In particular, generally speaking, the image of an ${\mathfrak{X}}$-maximal subgroup is not ${\mathfrak{X}}$-maximal in the image of a homomorphism. Nevertheless, there are group homomorphism…
Let $\boldsymbol{G}$ be an algebraic group of exceptional Lie type in characteristic $p$, $G=\boldsymbol{G}^σ$ its fixed-point subgroup under the action of a Steinberg endomorphism $σ$, and $\overline{G}$ an almost simple group with socle $G$. A maximal subgroup $M<\overline{G}$ is called non-generic if it is almost simple and its socle is not isomorphic to a group of Lie type in characteristic $p$. A finite subgroup $H$ of $\boldsymbol{G}$ is Lie primitive if it does not lie in any proper closed positive-dimensional subgroup of $\boldsymbol{G}$; it is Lie imprimitive if it lies in a positive-…
In this paper we study the groups all whose maximal or all Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product the with generalizations of the Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We prove that a hereditary formation $\mathfrak{F}$ contains every group all whose Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product with $\mathrm{F}^*(G)$ if and only if $\mathfrak{F}$ is the class of all $σ$-nilpotent groups for some partition $σ$ of the set of all primes. We obtain a new characterization of the $σ$-nilpotent hypercenter, i.e. the $\mathfrak{F…
Unipotent normal subgroups of algebraic groups
本文研究了代数群的单势正则子群的性质，并探讨了其在光滑连通仿射群中的类比关系。
A subgroup $H$ of a group $G$ is called $\mathbb P$-subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset ... \subset H_n=G$ such that $|H_i:H_{i-1}|$ is a prime for all $i$. In this paper, we study the groups in which all primary cyclic subgroups are $\mathbb P$-subnormal.
Let $H$ be a subgroup of a group $G$. The permutizer $P_G(H)$ is the subgroup generated by all cyclic subgroups of $G$ which permute with $H$. A subgroup $H$ of a group $G$ is strongly permutable in $G$ if $P_U(H)=U$ for every subgroup $U$ of $G$ such that~$H\le U\le G$. We investigate groups with $\mathbb{P}$-subnormal or strongly permutable Sylow and primary cyclic subgroups. In particular, we prove that groups with all strongly permutable primary cyclic subgroups are supersoluble.
The motivating question of this paper is roughly the following: given a group scheme $G$ over $\mathbb{Z}_p$, $p$ prime, with semisimple generic fiber $G_{\mathbb{Q}_p}$, how far are open subgroups of $G(\mathbb{Z}_p)$ from subgroups of the form $X(\mathbb{Z}_p)\mathbf{K}_p(p^n)$, where $X$ is a subgroup scheme of $G$ and $\mathbf{K}_p(p^n)$ is the principal congruence subgroup $\operatorname{Ker} (G(\mathbb{Z}_p)\rightarrow G(\mathbb{Z}/p^n\mathbb{Z}))$? More precisely, we will show that for $G_{\mathbb{Q}_p}$ simply connected there exist constants $J\ge1$ and $\varepsilon>0$, depending only …
Let $G$ be a Garside group endowed with the generating set $\mathcal{S}$ of non-trivial simple elements, and let $H$ be a parabolic subgroup of $G$. We determine a transversal $T$ of $H$ in $G$ such that each $θ\in T$ is of minimal length in its right-coset, $H θ$, for the word length with respect to $\mathcal{S}$. We show that there exists a regular language $L$ on $\mathcal{S} \cup \mathcal{S}^{-1}$ and a bijection $\mathrm{ev} : L \to T$ satisfying $\mathrm{lg} (U) = \mathrm{lg}_\mathcal{S}( \mathrm{ev}(U))$ for all $U \in L$. From this we deduce that the coset growth series of $H$ in $G$ i…
Let $G$ be a locally compact topological group, $G_0$ the connected component of its identity element, and comp(G) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a topological group) by finitely many elements is generated (as a topological group) by a single element. The space SUB(G) of all closed subgroups of $G$ carries a compact Hausdorff topology called the Chabauty topology. Let $F_1(G)$, respectively, $R_1(G)$, denote the subspace of all discrete subgroups isomorphic to $\mathbb Z$, respectively, all subgroups i…
We present a proof of the following claim. Suppose that $n$ is an integer such that $n>1$ and that $k$ is any field. Suppose that $g$ is an element of $\mathrm{SL}(n,k)$ of infinite order. Then the set $\{h\in\mathrm{SL}(n,k)\mid <g,h>$ is a free group of rank two$\}$ is a Zariski dense subset of $\mathrm{SL}(n,\bar{k})$ where $\bar{k}$ is an algebraic closure of $k$.
We study 3-dimensional Poincaré duality pro-$p$ groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-$p$ group $G$ has a nontrivial finitely presented subnormal subgroup of infinite index, then either the subgroup is cyclic and normal, or the subgroup is cyclic and the group is polycyclic, or the subgroup is Demushkin and normal in an open subgroup of $G$. Also, we describe the centralizers of finitely generated subgroups of 3-dimensional Poincaré duality pro-$p$ groups.
Locally Solvable Radicals via Wilson Radical Sets and Subgroup Lattices
本文研究群的局部可解根，利用Wilson序列证明局部（可解-有限）群等的相关根等式，给出根成员的共轭、换位子刻画及近模子群格群的对应结果。
Let $G$ be a linear Lie group that acts on it's Lie algebra $\mathfrak{g}$ by the adjoint action: $\mathrm{Ad}(g)X=gXg^{-1}$. An element $X\in \mathfrak {g}$ is called $\mathrm{Ad}_G$-real if $-X = \mathrm{Ad}(g)X $ for some $g\in G$. An $\mathrm{Ad}_G$-real element $X$ is called strongly $\mathrm{Ad}_G $-real if $-X = \mathrm{Ad}(τ) X $ for some involution $τ\in G$. Let $K=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Let $\mathrm{U}_n(K)$ be the group of unipotent upper-triangular matrices over $K$. Let $\mathfrak{u}_n (K)$ be the Lie algebra of $\mathrm{U}_n(K)$ that consists of $n \times n$ u…
A note on characterized and statistically characterized subgroups of $\mathbb{T}=\mathbb{R}/\mathbb{Z}$
本文针对圆群的特征化子群与统计特征化子群问题，给出了比已有结果更简短的完全不同的证明。
We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$, analyzing the subnormal structure of $H$, and the orbit-stabilizer problem for $H$. Effective computation with subgroups of $\mathrm{GL}(n,\mathbb{Z}_m)$ is vital to this work. All algorithms have been implemented in GAP.
We prove that for any finite index subgroup $\Ga$ in $SL_n(\mathbb{Z})$, there exists $k=k(n)\in\mathbb{N}$, $\ep=\ep(\Ga)>0$, and an infinite family of finite index subgroups in $\Ga$ with a Kazhdan constant greater than $\ep$ with respect to a generating set of order $k$. On the other hand, we prove that for any finite index subgroup $\Ga$ of $SL_n(\mathbb{Z})$, and for any $\ep>0$ and $k\in \mathbb{N}$, there exists a finite index subgroup $\Ga'\leq \Ga$ such that the Kazhdan constant of any finite index subgroup in $\Ga'$ is less than $\ep$, with respect to any generating set of order $k$.…
We parametrize the set of irreducible characters of the Sylow $p$-subgroups of the Chevalley groups $\mathrm{D}_6(q)$ and $\mathrm{E}_6(q)$, for an arbitrary power $q$ of any prime $p$. In particular, we establish that the parametrization is uniform for $p \ge 3$ in type $\mathrm{D}_6$ and for $p \ge 5$ in type $\mathrm{E}_6$, while the prime $2$ in type $\mathrm{D}_6$ and the primes $2,$ $3$ in type $\mathrm{E}_6$ yield character degrees of the form $q^m/p^i$ which force a departure from the generic situations. Also for the first time in our analysis we see a family of irreducible characters …
We say that a subset $X$ quasi-isometrically boundedly generates a finitely generated group $Γ$ if each element $γ$ of a finite-index subgroup of $Γ$ can be written as a product $γ= x_1 x_2 \cdots x_r$ of a bounded number of elements of $X$, such that the word length of each $x_i$ is bounded by a constant times the word length of $γ$. A. Lubotzky, S. Mozes, and M.S. Raghunathan observed in 1993 that ${\rm SL}(n,{\mathbb Z})$ is quasi-isometrically boundedly generated by the elements of its natural ${\rm SL}(2,{\mathbb Z})$ subgroups. We generalize (a slightly weakened version of) this by showi…
Lifting subgroups of $\mathrm{PSL}_2$ to $\mathrm{SL}_2$ over local fields
本文研究非阿基米德局部域上从 PSL₂(K) 到 SL₂(K) 的子群提升问题，证明无 2-挠的离散子群总能提升，并举例说明两个条件缺一不可，同时简要讨论表示提升。
A theorem of Farb and Handel asserts that for $N\ge 4$, the natural inclusion from $\mathrm{Out}(F_N)$ into its abstract commensurator is an isomorphism. We give a new proof of their result, which enables us to generalize it to the case where $N=3$. More generally, we give sufficient conditions on a subgroup $Γ$ of $\mathrm{Out}(F_N)$ ensuring that its abstract commensurator $\mathrm{Comm}(Γ)$ is isomorphic to its relative commensurator in $\mathrm{Out}(F_N)$. In particular, we prove that the abstract commensurator of the Torelli subgroup $\mathrm{IA}_N$ for all $N\ge 3$, or more generally any…
This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\mathrm{Out}(F_{\tt n})$, the outer automorphism group of a nonabelian free group $F_{\tt n}$. We prove that, for every subgroup $H$ of $\mathrm{Out}(F_{\tt n})$, there exists an element $ϕ\in H$ such that, for every element $g$ of $F_{\tt n}$, the conjugacy class $[g]$ has polynomial growth under iteration of $ϕ$ if and only if $[g]$ has polynomial growth under iteration of every element of $H$.
Let $\Bbb P$ be the set of all primes. A subgroup $H$ of a group $G$ is called {\it $\mathbb P$-subnormal} in $G$, if either $H=G$, or there exists a chain of subgroups $H=H_0\le H_1\le \ldots \le H_n=G, \ |H_{i}:H_{i-1}|\in \Bbb P, \ \forall i.$ A group $G$ is called {\it widely supersoluble}, $\mathrm{w}$-supersoluble for short, if every Sylow subgroup of $G$ is $\mathbb P$-subnormal in $G$. A group $G=AB$ with $\mathbb P$-subnormal $\mathrm{w}$-supersoluble subgroups $A$ and $B$ is studied. The structure of its $\mathrm{w}$-supersoluble residual is obtained. In particular, it coincides with…
Exponentially distorted subgroups in wreath products
本文证明在两类圈积群 $\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$ 与 $\mathbb{Z} \wr F_2$ 中均存在指数扭曲的子群。
Let $V$ be a vector space over the field of order $2$. We investigate subgroups of the linear group $GL(V)$ which are generated by a conjugacy class $D$ of elements of order $3$ such that all $d$ in $D$ have $2$-dimensional commutator space $[V,d]$.
We develop a theory of Ennola duality for subgroups of finite groups of Lie type, relating subgroups of twisted and untwisted groups of the same type. Roughly speaking, one finds that subgroups $H$ of $\mathrm{GU}_d(q)$ correspond to subgroups of $\mathrm{GL}_d(-q)$, where $-q$ is interpreted modulo $|H|$. Analogous results for types other than $\mathrm A$ are established, including for exceptional types where the maximal subgroups are known, although the result for type $\mathrm D$ is still conjectural. Let $M$ denote the Gram matrix of a non-zero orthogonal form for a real, irreducible repre…
Let $N \geq 2$ and let $\mathrm{Out}(F_N)$ be the outer automorphism group of a nonabelian free group of rank $N$. Let $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ be the finite index subgroup of $\mathrm{Out}(F_N)$ which is the kernel of the natural action of $\mathrm{Out}(F_N)$ on $H_1(F_N,\mathbb{Z}/3\mathbb{Z})$. We show that $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ is an $R$-group, that is, for every $ϕ,ψ\in \mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$, if there exists $k \in \mathbb{N}^*$ such that $ϕ^k=ψ^k$, then $ϕ=ψ$. This answers a question of Handel and Mosher. We then use the fact that $\mathr…
Let $Γ< \mathrm{GL}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{Sub}(Γ)$ the space of all subgroups of $Γ$ with the, compact, metric, Chabauty topology. An invariant random subgroup (IRS) of $Γ$ is a conjugation invariant Borel probability measure on $\mathrm{Sub}(Γ)$. An $\mathrm{IRS}$ is called nontrivial if it does not have an atom in the trivial group, i.e. if it is nontrivial almost surely. We denote by $\mathrm{IRS}^{0}(Γ)$ the collection of all nontrivial $\mathrm{IRS}$ on $Γ$. We show that there exits a free subgroup $F < Γ$ and a…
In [Ann. Math. 190 (2019), 657-661], James Hyde presented the first example of non-left-orderable, finitely generated subgroup of $\mathrm{Homeo}(D,\partial D)$, the group of homeomorphisms of the disk fixing the boundary. This implies that the group $\mathrm{Homeo}(D,\partial D)$ itself is not left-orderable. We revisit the construction, and present a slightly different proof of purely dynamical flavor, avoiding direct references to properties of left-orders. Our approach allows to solve the analogue problem for actions on the circle.
On medium-rank Lie primitive and maximal subgroups of exceptional groups of Lie type
本文研究特征p李型群H嵌入例外代数群G的李本原子群，排除PSL_2型，证明除少数例外（如E_8中的若干小群）外不存在，并推广到自同构稳定性和极大子群刻画，结合表示论与计算方法。
On varieties where $\mathrm{CS}\mathfrak{X}$ implies $\mathfrak{X}\mathrm{T}$
本文研究满足CSX蕴含XT的群簇，给出新的簇实例，其中所有极大X-子群malnormal的群满足任意两个交非平凡的X-子群生成X-子群。
On the intersection of $\mathfrak{F}$-maximal subgroups of a finite group
本文研究了有限群G的所有F-极大子群交集Int_F(G)的性质，证明了Int_F(G/Int_F(G))≈1当且仅当F包含所有其所有F-子群均为F-子正规的群，推导了Skiba和Beidleman等人的相关结论。
Lie type quotients of the maximal unipotent subgroup of Kac-Moody groups of type $\mathrm{HB}_{2}^{(2)}$
本文构造了秩无界的 Lie 型有限单群无限族作为类型 $\mathrm{HB}_{2}^{(2)}$ 的 Kac-Moody 群极大幂幺子群的商，并据此首次构建出由无界秩 Lie 型群生成的高维扩展图无限族。
Factorization number and subgroup commutativity degree via spectral invariants
本文利用子群不可置换图的邻接矩阵与拉普拉斯矩阵建立新的组合公式，从而用谱不变量表示有限群的因子分解数和子群交换度。
We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) Möbius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of $\mathrm{SL}(2, R)$ for a localization $R= \mathbb{Z}[\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a Möbius subgroup $G$ is not free by showing that it has finite index in the relevant $\mathrm{SL}(2, R)$. Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index …
By Cartan's Theorem, every closed subgroup $H$ of a real (or $p$-adic) Lie group $G$ is a Lie subgroup. For Lie groups over a local field ${\mathbb K}$ of positive characteristic, the analogous conclusion is known to be wrong. We show more: There exists a ${\mathbb K}$-analytic Lie group $G$ and a non-discrete, compact subgroup $H$ such that, for every ${\mathbb K}$-analytic manifold $M$, every ${\mathbb K}$-analytic map $f\colon M\to G$ with $f(M)\subseteq H$ is locally constant. In particular, the set $H$ does not admit a non-discrete ${\mathbb K}$-analytic manifold structure which makes the…
A closure operator on the subgroup lattice of $\mathrm{GL}(n,q)$ and $\mathrm{PGL}(n,q)$ in relation to the zeros of the Möbius function
本文在PGL(V)和GL(V)的子群格上定义闭包算子，用以刻画Möbius函数非零的子群，并对满足特定不变子空间格条件的指数为m的闭子群个数建立了仅依赖于m的多项式上界。
We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z} \rtimes ... \rtimes \mathbb{Z})$. Then for all $k \geq 2$, we calculate $m_n(G_k)$, the number of maximal subgroups of $G_k$ of index $n$, exactly. Also, for infinitely many groups $H_k$ of the form $\mathbb{Z}^2 \rtimes G_2$, we calculate $m_n(H_k)$ exactly.
On $\mathrm{K}\mathfrak F$-subnormality and submodularity in a finite group
本文研究有限群中素子群的子模性与 $\mathrm{K}\mathfrak U_1$-次正规性的等价关系，并证明可解 $\mathrm{K}\mathfrak{F}$-次正规子群必含于可解根。
Normal closure of finite subgroups of $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$
本文研究了有限子群在$\mathrm{Aut}(F_n)$和$\mathrm{Out}(F_n)$中的正规闭包性质，得出了关于其结构和闭包的结论。
For a sequence $\{U_n\}_{n = 1}^\infty$ of finite index subgroups of a direct product $G = A \times B$ of finitely generated groups, we show that $$\lim_{n \to \infty} \frac{\min\{|X| : \langle X \rangle = U_n\}}{[G : U_n]} = 0$$ once $[A : A \cap U_n], [B : B \cap U_n] \to \infty$ as $n \to \infty$. Our proof relies on the classification of finite simple groups. For $A,B$ that are finitely presented we show that $$ \lim_{n \to \infty} \frac{\log |\mathrm{Torsion}(U_n^{\mathrm{ab}})|}{[G : U_n]} = 0. $$
In this paper, we characterize the finite groups $G$ of even order with the property that for any involution $x$ and element $y$ of $G$, $\langle x, y \rangle$ is isomorphic to one of the following groups: $\mathbb{Z}_2,$ $\mathbb{Z}_2^2$, $\mathbb{Z}_4$, $\mathbb{Z}_6$, $\mathbb{Z}_2 \times \mathbb{Z}_4$, $\mathbb{Z}_2 \times\mathbb{Z}_6$ and $A_4$. As a result, a characterization will be obtained for the finite groups all of whose Cayley graphs of degree $3$ have integral spectrum.
We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $\mathrm{FP}_\infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.
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