Luis Jorge Sánchez Saldaña
We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for $\underline{E}G$ and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one relator groups, the Hilbert modular group and $3$-manifold groups.
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A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$Π$-property in $G$ if $G$ has a chief series $Γ_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant n)$ of $Γ_{G}$, either $(H\cap G_{i})G_{i-1}/G_{i-1}$ is a Sylow $p$-subgroup of $G_{i}/G_{i-1}$ or $|G/G_{i-1}: N_{G/G_{i-1}}((H\cap G_{i})G_{i-1}/G_{i-1})|$ is a $p$-number. In this paper, we mainly investigate the structure of finite groups with some primary subgroups satisfying partial $S$-$Π$-property.
A finite group $G$ is said to satisfy $C_π$ for a set of primes $π$, if $G$ possesses exactly one class of conjugate $π$-Hall subgroups. In the paper we obtain a criterion for a finite group $G$ to satisfy $C_π$ in terms of a normal series of the group.
In this note we describe the finite groups $G$ having $|G|-2$ cyclic subgroups. This partially solves the open problem in the end of \cite{3}.
Finite groups with some subgroups satisfying the partial $ Π$-property
本文提出了子群在有限群中满足部分 $Π$-性质的定义，并研究了满足该性质的某些子群对有限群结构的影响。
Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal elementary abelian Sylow $p$-subgroup $P$ such that $p^d<|P|$ in which all subgroups of order $p^d$ are complemented.
In this paper we consider the {\em conjugacy stability} property of subgroups and provide effective procedures to solve the problem in several classes of groups. In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable. Then we further generalize this result to quasi-convex subgroups of torsion-free hyperbolic groups and finitely generated subgroups of limit groups.
For a non-cyclic finite group $X$ let $σ(X)$ be the least number of proper subgroups of $X$ whose union is $X$. Precise formulas or estimates are given for $σ(S \wr C_{m})$ for certain nonabelian finite simple groups $S$ where $C_m$ is a cyclic group of order $m$.
Given a finite non-cyclic group $G$, call $σ(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $σ(G)$ for $G$ a group with a unique minimal normal subgroup $N$ isomorphic to $A_n^m$ where $n \geq 5$ and $G/N$ is cyclic. We also show that $σ(A_5 \wr C_2)=57$.
Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups.
Finite groups with some subgroups of prime power order satisfying the partial $ Π$-property
该文提出有限群子群的部分$Π$-性质的定义，研究若干素数幂阶子群满足该性质时有限群$G$的结构，属于群论领域的结构研究。
Solvable Supplements to Normalizers of Cyclic 2-Subgroups
本文放宽了可解补充条件至仅对循环2-子群成立，刻画了满足该条件的有限群的非交换合成因子为特定PSL2(q)族，并给出可解性的充要条件。
We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and …
We give an explicit characterization of which direct products $G$ of surface groups of Euclidean type satisfy that the fixed subgroup of any automorphism (or endomorphism) of $G$ is compressed, and of which is it always inert.
In this note we study a class of finite groups for which the orders of subgroups satisfy a certain inequality. In particular, characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_2$ and $S_3$ are obtained.
Finite groups with dense solitary subgroups
本研究针对具有稠密孤立子群的有限群展开，明确了子群格非空开区间含孤立子群的群定义，并找出所有满足该性质的有限群，填补了相关群论研究的空白。
Suppose that $G$ is a group, $H$ and $K$ are proper isomorphic central subgroups of $G$, and $\mathfrak{G}$ is an HNN-extension of $G$ with the associated subgroups $H$ and $K$. We prove necessary and sufficient conditions for $\mathfrak{G}$ to be residually a $\mathcal{C}$-group, where $\mathcal{C}$ is a class of groups closed under taking subgroups, extensions, homomorphic images, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for each $y \in Y$.
We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type G2, focusing on the case of number fields and p-adic fields. In the same context we characterize the rational conjugacy classes of A2 subgroups of G2. Having obtained the concrete characterization, we then relate it to the more abstract characterization which can be given in terms of Galois cohomology. We note that these results on A2 subgroups were simultaneously and independently developed in the work of Hooda whereas the results on tori were simultaneously and independently developed in t…
We show that the outer automorphism groups of graph products of finitely generated abelian groups satisfy the Tits alternative, are residually finite, their so-called Torelli subgroups are finitely generated, and they satisfy a dichotomy between being virtually nilpotent and containing a non-abelian free subgroup that is determined by a graphical condition on the underlying labelled graph. Graph products of finitely generated abelian groups simultaneously generalize right-angled Artin groups (RAAGs) and right-angled Coxter groups (RACGs), providing a common framework for studying these groups.…
Finite groups with simple or $p$-nilpotent maximal subgroups
研究《库罗夫卡笔记本》中关于极大子群的问题，对于奇数\(p\)，证明满足特定条件的非单非可解群是\(p\)-幂零群和\(p\)-可分解群，\(p = 2\)时改进相关结果，给出特定非可解群的完整分类。
Let $F_{2,d}$ denote the free class-2-nilpotent group on $d$ generators. We compute the normal zeta functions $ζ^\triangleleft_{F_{2,d}}(s)$, prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders.
A valuated group with normal forms is a group with an integer-valued length function satisfying some Lyndon's axioms and an additional axiom considered by Hurley. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko-Neumann's theorem. We study also the CSA property in such groups.
In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |π(G)|+2 if and only if G is isomorphic to A5, where n is the number of isomorphism classes of derived subgroups of G and π(G) is the set of prime divisors of the group G.
In this article we introduce and study a class of finite groups for which the orders of normal subgroups satisfy a certain inequality. It is closely connected to some well-known arithmetic classes of natural numbers.
Let p be a prime. We classify finitely generated pro-p groups G which satisfy d(H) = d(G) for all open subgroups H of G. Here d(H) denotes the minimal number of topological generators for the subgroup H. Within the category of p-adic analytic pro-p groups, this answers a question of Iwasawa.
We describe an algorithm that computes the index of a finitely generated subgroup in a finitely $L$-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely $L$-presented group is decidable. As an application, we consider the low-index subgroups of some self-similar groups including the Grigorchuk group, the twisted twin of the Grigorchuk group, the Grigorchuk super-group and the Hanoi 3-group
Alexander's theorem for stabilizer subgroups of Thompson's group
本文证明Thompson群$F$在单位区间自然作用下的几乎所有稳定子群都满足Alexander定理，即任何纽结或链环均可由这些子群的元素构造。
The Sylow p-subgroups of the symmetric group S_p^n satisfy the appropriate generalization of Maschke's Theorem to the case of a p'-group acting on a (not necessarily abelian) p-group. Moreover, some known results about the Sylow p-subgroups of S_p^n are stated in a form that is true for all primes p.
Let S be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph S, which we call canonical parabolic subgroups. A natural extension of the definition leads to canonical quasiparabolic subgroups. It is shown that the centralisers of subsets of G are the conjugates of canonical quasiparabolic centralisers satisfying certain graph theoretic conditions.
We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.
Finite groups with dense ${\cal CD}$-subgroups
本文定义并研究子群格每个非空开区间均含 Chermak–Delgado 格成员的有限群，即具有稠密 ${\cal CD}$-子群的有限群。
We generalise a technical tool, originally developed by Pervova for the study of maximal subgroups in Grigorchuk and GGS groups, to all weakly branch groups satisfying a natural condition, and in particular to all branch groups. We then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group. As a further application of this result, we show that every maximal subgroup of the Basilica group is of finite index.
We prove that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property we show that almost all finite subsets of the group generate free subgroups. We derive consequences of these theorems on Thompson's group $F$, weakly branch groups, automorphism groups of regular trees and profinite groups with alternating composition factors of unbounded degree.
Let $H$ be a subgroup of a group $G$. $H$ is said satisfying $Π$-property in $G$, if $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $π(HK/K\cap L/K))$-number for any chief factor $L/K$ of $G$, and, if there is a subnormal supplement $T$ of $H$ in $G$ such that $H\cap T\le I\le H$ for some subgroup $I$ satisfying $Π$-property in $G$, then $H$ is said $Π$-normal in $G$. By these properties of some subgroups, we obtain some new criterions of $p$-nilpotency of finite groups.
On the non-existence of finite groups with certain normal subgroups
该研究证明不存在同时满足两个正规子群同构且商群不同构的有限群。
We study the class of finite groups $G$ satisfying $Φ(G/N)= Φ(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
Let C be the centralizer in a finite Weyl group of an elementary abelian 2-subgroup. We show that every complex representation of C can be realized over the field of rational numbers. The same holds for a Sylow 2-subgroup of C.
Statistically characterized subgroups related to arithmetic-type sequence of integers
本文研究了更广泛的算术型整数序列相关的统计子群，总结了其在算术序列和某些非算术序列中的特殊性质，并展示了该类别的显著不同行为。
For a finite group $G$ we investigate the difference between the maximum size MaxDim$(G)$ of an "independent" family of maximal subgroups of $G$ and maximum size $m(G)$ of an irredundant sequence of generators of $G$. We prove that MaxDim$(G)=m(G)$ if the derived subgroup of $G$ is nilpotent. However MaxDim$(G)-m(G)$ can be arbitrarily large: for any odd prime $p,$ we construct a finite soluble group with Fitting length 2 satisfying $m(G)=3$ and MaxDim$(G)=p.$
Fix a set of primes $π$. A finite group is said to satisfy $C_π$ or, in other words, to be a $C_π$-group, if it possesses exactly one class of conjugate $π$-Hall subgroups. The pronormality of $π$-Hall subgroups in $C_π$-groups is proven, or, equivalently, we prove that $C_π$ is inherited by overgroups of $π$-Hall subgroups. Thus an affirmative solution to Problem 17.44(a) from the "Kourovka notebook" is obtained. We also provide an example, showing that Hall subgroups in finite groups are not pronormal in general.
We determine the combinatorial depth of certain subgroups of simple Suzuki groups Sz(q), among others the depth of their maximal subgroups. We apply these results to determine the ordinary depth of these subgroups.
We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are "optimal" in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any $n\geq2^{48}$ such that $n$ is either odd or divisible by $2^9$, any infinite free Burnside group with exponent $n$ is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.
Let $ n, q $ be positive integers. We show that if $ G $ is a finitely generated residually finite group satisfying the identity $ [x,_ny^q]\equiv 1, $ then there exists a function $ f(n) $ such that $ G $ has a nilpotent subgroup of finite index of class at most $ f(n) $. We also extend this result to locally graded groups.
Groups satisfying Gaschütz's complement theorem
该研究刻画了使Gaschütz补定理在去掉阿贝尔假设后仍成立的有限群N，补充了第二作者的前期工作。
We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.
Groups with Finiteness Conditions on commutator subgroups
本文研究了特定换位子群集合为有限的群的结构，重点分析了这类群与具有有限导出子群的群之间的关系。
We improve the existing lower bounds on the order of counterexamples to a conjecture by P. Schmid, determine some properties of the possible counterexamples of minimum order for each prime, and the isomorphism type of the center of the Frattini subgroup for the counterexamples of order 256. We also show that nonabelian metacyclic p-groups, nonabelian groups of maximal nilpotency class and 2-groups of coclass two satisfy the conjecture.
For a discrete group $Γ$ satisfying some finiteness conditions we give a Bredon projective resolution of the trivial module in terms of projective covers of the chain complex associated to certain posets of subgroups. We use this to give new results on dimensions of $Eγ$ and to reprove that for virtually solvable groups, $\underline{\cd}Γ=\vcdΓ$. We also deduce a formula to compute the Euler class of $Eγ$ for $Γ$ virtually solvable of type $\FP_\infty$ and use it to compute orbifold Euler characteristics.
Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $α(G)=\frac{|L_1(G)|}{|G|}$. The aim of this paper is to study the class $\cal{C}$ of finite nilpotent groups having $α(G)=\frac{3}{4}$. We show that if $G$ belongs to this class, then it is a 2-group satisfying certain conditions. Also, we study the appartenance of some classes of finite groups to $\cal{C}$.
Certain residual properties of HNN-extensions with normal associated subgroups
本文研究了具有正规关联子群的HNN-扩张，证明了其剩余为特定群类的充分必要条件，并给出了关于剩余有限性和剩余p-有限性的新事实。
Supercharacter theory is developed by P. Diaconis and I. M. Isaacs as a natural generalization of the classical ordinary character theory. Some classical sums of number theory appear as supercharacters which are obtained by the action of certain subgroups of GLd(Zn) on Zdn. In this paper we take Zdp , p prime, and by the action of certain subgroups of GLd(Zp) we find supercharacter table of Zdp .
Two-Generator Discrete and Faithful Subgroups of Tree Automorphisms
本文部分分类了三叉树自同构群中满足小几何量限制的二生成元离散子群，并放宽限制时构造了无限族群图。
Some Properties of normal subgroups determined from character tables
本文分析了可从G特征标表确定的正规子群结构性质，扩展了Thompson定理和Taketa定理，并引入了幂零类受不可约特征标度数量界定的新类nMI群。
This paper proves under certain conditions the existence of an algorithm, which detects relatively quasiconvex subgroups $H$ of relatively hyperbolic groups $(G,\mathbb{P})$. Additionally, this algorithm outputs an induced peripheral structure of $(G,\mathbb{P})$ on $H$.
We classify closed abelian subgroups of the simple groups $G_2$, $F_4$, $Aut(so(8))$ having centralizer the same dimension as the dimension of the subgroup, as well as finite abelian subgroups of certain spin and half-spin groups having finite centralizer.
We study special subgroups of infinite groups that generalize double centralizers. We analyze sufficient conditions for descending chains of such subgroups to stop after finitely many steps. We discuss whether this phenomenon can happen in the class of groups satisfying chain condition on centralizers.
We solve an open problem of Herfort and Ribes: Profinite Frobenius groups of certain type do occur as closed subgroups of free profinite products of two profinite groups. This also solves a question of Pop about prosolvable subgroups of free profinite products.
A subgroup $H$ of a finite group $G$ is said to satisfy $Π$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $π(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $Π$-supplemented in $G$ if there exists a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq I\leq H$, where $I$ satisfies $Π$-property in $G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that some primary subgroups of $G$ are $Π$-supplemented in $G$. The main result we proved improves a large number of earlier results.
Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion which involves root systems.
Stable subgroups of graph products
研究图积中稳定子群的性质，证明其等价于在关联接触图中准等距嵌入的子群，并推广了Koberda等人关于稳定与纯洛希多里子群等价的结果。
Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $χ(g)\neq 0$ for all irreducible characters $χ$ of $G$. Such an element is said to be non-vanishing in $G$. Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property, then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.
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