Marius Tarnauceanu
In this paper we introduce and study the poset of equivalence classes of subgroups of a finite group $G$, induced by the isomorphism relation. This contains the well-known lattice of solitary subgroups of $G$. We prove that in several particular cases it determines the structure of $G$.
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In this note, we determine the finite groups whose poset of conjugacy classes of subgroups has breaking points. This leads to a new characterization of the generalized quaternion $2$-groups. A generalization of this property is also studied.
We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index, and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.
On Posets of Classes of Subgroups with Same Set of Orders of Elements
本文研究了有限群中具有相同元素阶集的子群类的偏序集，证明其在p群中为链，并探讨了该偏序集在循环群和二面体群中的格结构及其分配和模特性。
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.
On Posets of Classes of Automorphic Subgroups of Finite Groups
本文研究了有限群的自同构子群类偏序集AutCl(G)，通过引入偏序解决了Tarnauceanu提出的问题5，并证明了AutCl(Dn)和AutCl(Q4m)是分配格，同时刻画了使AutCl(G)构成链的所有有限群类。
We study the connectivity of the coset poset and the subgroup poset of a group, focusing in particular on simple connectivity. The coset poset was recently introduced by K. S. Brown in connection with the probabilistic zeta function of a group. We further Brown's study of the homotopy type of the coset poset, and in particular generalize his results on direct products and classify direct products with simply connected coset posets. The homotopy type of the subgroup poset L(G) has been examined previously by Kratzer, Thevenaz, and Shareshian. We generalize some results of Kratzer and Thevenaz, …
We consider ordered tuples in finite groups generating nilpotent subgroups. Given an integer $q$ we consider the poset of nilpotent subgroups of class less than $q$ and its corresponding coset poset. These posets give rise to a family of finite Dirichlet series parametrized by the nilpotency class of the subgroups, which in turn reflect probabilistic and topological invariants determined by these subgroups. Connections of these series to filtrations of classifying spaces of a group are discussed.
An affirmative answer to a question on connectivity of p-subgroup posets with irreducible characters
本文通过研究p-子群格中的不可约特征标问题，证明了关于子群交集与连通分量的不等式，从而回答了Meng和Yang提出的问题。
We study some properties of the coset poset associated with the family of subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is $2$-Engel, and $2$-connected if and only if the group is nilpotent of class $2$ or less. We determine the homotopy type of the coset poset for the group of $4\times 4$ upper unitriangular matrices over $\mathbb{F}_p$, and for the Burnside groups of exponent $3$.
We study the homotopy properties of the posets of p-subgroups Sp(G) and Ap(G) of a finite group G, viewed as finite topological spaces. We answer a question raised by R.E. Stong in 1984 about the relationship between the contractibility of the finite space Ap(G) and that of Sp(G) negatively, and describe the contractibility of Ap(G) in terms of algebraic properties of the group G.
The Chermak-Delgado Measure as a Map on Posets
本文研究有限群的 Chermak-Delgado 度量作为偏序集映射的条件，给出了其为单调函数的充要条件，并探讨了限制在中心化子上的递增性。
We prove that the Monster does not contain any subgroup isomorphic to PSL_2(27).
This paper studies when a pair of elements in F are the images of the standard generators of F under a self monomorphism.
Let $G$ be a finite group and $N\unlhd G$. In this note, we construct a class poset of $G$ for some cyclic subgroup $C$ of $G$. And we find a relation between $m_{G,N}$ and the Euler characteristic of some nerve spaces of these posets(see Main Theorem).
In this paper, we focus on a question of M. Newman on isomorphic subgroups of solvable groups. We get a reduction theorem of this question: for each prime q, assume that this question holds for every characteristic q-groups, then this question holds for every finite solvable groups. Using this reduction theorem, we get some partial answers about this question.
The pro-isomorphic zeta function of a torsion-free finitely generated nilpotent group G enumerates finite index subgroups H such that H and G have isomorphic profinite completions. It admits an Euler product decomposition, indexed by the rational primes. We manufacture the first example of a torsion-free finitely generated nilpotent group G such that the local Euler factors of its pro-isomorphic zeta function do not satisfy functional equations. The group G has nilpotency class 4 and Hirsch length 25. It is obtained, via the Malcev correspondence, from a Z-Lie lattice L with a suitable algebra…
Connectivity of $p$-subgroup posets with irreducible characters
本文研究了有限群中p-子群与不可约特征标之间的连通性，证明了特定条件下偏序集的连通分量结构。
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…
M.Newman has asked if it is the case that whenever H and K are isomorphic subgroups of a finite solvable group G with H maximal, then K is also maximal. This question was considered in a paper of I.M. Isaacs and the second author, where (among other things) the answer was shown to be affirmative if H has an Abelian Sylow 2-subgroup. Here, we show that the answer is affirmative unless the index of H is a power of a prime less than 5 and we obtain further restrictions on the structure of a purported minimal counterexample.
A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. The groups $\operatorname{Aut}(A)$ of such posets $A$ have a natural topology in which $\operatorname{Aut}(A)$ are Polish topological groups. We consider the problem whether $\operatorname{Aut}(A)$ contains a dense free subgroup of two generators. We show that if $A$ is ultrahomogeneous, then $\operatorname{Aut}(A)$ contains such subgroup. Moreover, we characterize whose countable ultrahomogeneous posets $A$ such that for each natural $m$, the set of all cyclically dense …
Classes of Subgroups of a finite group with isomorphic quotient groups
本文研究有限群中具有同构商群的子群类构成的偏序集QIso(G)，针对阿贝尔群及几类非阿贝尔亚循环群展开分析，得出QIso(G)的同构性质及相关群类的刻画结论。
Let G be a group. The intersection graph G(G) of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G; and there is an edge between two distinct vertices X and Y if and only if X \ Y 6= 1 where 1 denotes the trivial subgroup of G: It was conjectured in [2] that two (non cyclic) finite abelian groups with isomorphic intersection graphs are isomorphic. In this paper we study this conjecture and show that it is almost true. For any inite abelian group D let Dnc be the product of all noncyclic Sylow subgroup…
On subgroups of the Monster isomorphic to $PSL_2(8)$
本文通过计算机计算分类 Monster 中包含 $7B$-元素的 $PSL_2(8)$ 子群，证明其不存在，并验证了早期及独立计算的可复现性。
Let $P$ be a finite $p$-group and $p$ be an odd prime. Let $\mathcal{A}_p(P)_{\geq2}$ be a poset consisting of elementary abelian subgroups of rank at least 2. If the derived subgroup $P'\cong C_p\times C_p$, then the spheres occurring in $\mathcal{A}_p(P)_{\geq2}$ all have the same dimension.
In answer to a question of P. Hall, we supply another construction of a group which is isomorphic to each of its non-trivial normal subgroups.
Let $G$ be a finite group and $N\unlhd G$ with $|G: N|=p$ for some prime $p$. In this note, to compute $m_{G,N}$ directly, we construct a class poset $\mathfrak{T}_{C}(G)$ of $G$ for some cyclic subgroup $C$. And we find a relation between $m_{G,N}$ and the Euler characteristic of the nerve space $|N(\mathfrak{T}_{C}(G))|$ (see the Theorem 1.3). As an application, we compute $m_{S_5, A_5}=0$ directly, and get $S_5$ is a $B$-group.
We consider two variants of those Abelian groups with all proper characteristic subgroups isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups with all proper fully invariant subgroups isomorphic, which are studied in details by the current authors in Commun. Algebra (2015). In addition, we also examine those Abelian groups having at least one proper characteristic subgroup isomorphic to the whole group. The established by us results somewhat extend those obtained by Grinshpon-Nikolskaya in Tomsk State Univ. J. Math. …
We show that the order complex of the poset of all cosets of all proper subgroups of a finite group $G$ is never $\mathbb{F}_{2}$-acyclic and therefore never contractible. This settles a question of K. S. Brown.
On Abelian Groups Having Isomorphic Proper Strongly Invariant Subgroups
本文研究了所有真强不变子群同构的阿贝尔群的两种变体，深入探讨其基本与特殊性质，并与真全不变或特征子群同构的阿贝尔群进行对比，同时探索了真强不变子群与整个群同构的阿贝尔群。
A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to G. We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: First, we show any characteristic proper finite-index subgroup isomorphic to G arises by pulling back a finite-index subgroup of the abelianization, and secondly, we prove special cases (fo…
Higman--Thompson groups $F_n$ all the way down
证明对$n\geq2$，希格曼 - 汤普森群$F_n$有无限指数且同构于自身的极大子群。构造相关子群链，刻画了特定同胚，给出将共轭问题转化为算法的机制，并应用于解决一些群同构问题。
In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph $Γ(H)$. For a finitely generated group $G$ we consider the poset $P_{\text{fi}}(G)$ of all right cosets of all proper finite index subgroups of $G$. We use the theory of subgroup graphs to prove that for many finitely generated infinite groups the order complex $ΔP_{\text{fi}} (G)$ and the corresponding nerve complex are contractible.
This paper studies the poset of eigenspaces of elements of an imprimitive unitary reflection group, for a fixed eigenvalue, ordered by the reverse of inclusion. The study of this poset is suggested by the eigenspace theory of Springer and Lehrer. The posets are shown to be isomorphic to certain subposets of Dowling lattices (the `d-divisible, k-evenly coloured Dowling lattices'). This enables us to prove that these posets are Cohen-Macaulay, and to determine the dimension of their top homology.
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 $y \in Y$.
Topological concepts may be applied to any poset via the simplicial complex of finite chains. The coset poset C(G) of a finite group G (consisting of all cosets of all proper subgroups of G, ordered by inclusion) was introduced by Kenneth S. Brown in his study of the probabilistic zeta function P(G,s), and Brown's results motivate the study of the homotopy type of C(G). After a detailed introduction to basic simplicial topology and homotopy theory for finite posets (Chapter 2), we examine the connectivity of the coset poset (Chapter 3). Of particular interest is the fundamental group, and we p…
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 $\cal{C}$.
A real chain condition for groups
本文提出偏序集不含与实数集序同构子集的弱链条件，并证明将其用于不含正规性推广性质的子群偏序集，可统一完善广义根群中此前依赖更强链条件的结果。
The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\inπ(B)$ and $2\notinπ(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$ of a group $G$ is called {\it finite} if for all $ g\in G $ the subgroups $ \langle a, a^g \rangle $ are finite. In the paper, it is proved that the group with finite element of order $4$ and strongly embedded subgroup isomorphic to the Borel subgroup of $U_3(Q)$ over a locally finite field $Q$ of characteristic $2$ is locally finite and isomorphic to the group…
We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable. We also use this geometric perspective to recover and unify results describing the normalisers of para…
In this paper we produce many examples of thin subgroups of special linear groups that are isomorphic to the fundamental groups of non-arithmetic hyperbolic manifolds. Specifically, we show that the non-arithmetic lattices in $\mathrm{SO}(n,1)$ constructed by Gromov and Piatetski-Shapiro can be embedded into $\mathrm{SL}_{n+1}(\mathbb{R})$ so that their images are thin subgroups
We prove that each infinite 2-group with a unique 2-element subgroup is isomorphic either to the quasicyclic 2-group or to the infinite group of generalized quaternions.
In the note we prove that all composition factors of a finite group possessing a Carter subgroup of odd order either are abelain, or are isomorphic to $L_2(3^{2n+1})$.
Suppose that $(W,S)$ is a Coxeter system with associated Artin group $A$ and with a simplicial complex $L$ as its nerve. We define the notion of a "standard abelian subgroup" in $A$. The poset of such subgroups in $A$ is parameterized by the poset of simplices in a certain subdivision $L_\oslash$ of $L$. This complex of standard abelian subgroups is used to generalize an earlier result from the case of right-angled Artin groups to case of general Artin groups, by calculating, in many instances, the smallest dimension of a manifold model for $BA$. (This is the "action dimension" of $A$ denoted …
The authors classify the finite index subgroups of R. Thompson's group $F$. All such groups that are not isomorphic to $F$ are non-split extensions of finite cyclic groups by $F$. The classification describes precisely which finite index subgroups of $F$ are isomorphic to $F$, and also separates the isomorphism classes of the finite index subgroups of $F$ which are not isomorphic to $F$ from each other; characterizing the structure of the extensions using properties of the structure of the finite index subgroups of $Z\times Z$.
In this paper, we investigate the M{ö}bius function $μ\_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. We introduce and develop a new approach to study $μ\_{\mathcal{S}}$ by using the Hilbert series of $\mathcal{S}$. The latter enables us to provide formulas for $μ\_{\mathcal{S}}$ when $\mathcal{S}$ belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given.
Assume that a finite almost simple group with simple socle isomorphic to an exceptional group of Lie type possesses a solvable Hall subgroup. Then there exist four conjugates of the subgroup such that their intersection is trivial.
Invariable generation of finite simple groups and rational homology of coset posets
本文证明每个有限单群都可由一个西洛子群与一个循环群不变生成，并据此推出任意有限群的陪集偏序集阶复形具有非平凡约化有理同调。
The multiplier representation of the generalized symmetry group of a quasiperiodic flow on the n-torus defines, for each subgroup of the multiplier group of the flow, a group invariant of the smooth conjugacy class of that flow. This group invariant is the internal semidirect product of a subgroup isomorphic to the n-torus by a subgroup isomorphic to that subgroup of the multiplier group. Each subgroup of the multiplier group is a multiplicative group of algebraic integers of degree at most n, which group is isomorphic to an abelian group of n by n unimodular matrices.
Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every …
Subgroups with all finite lifts isomorphic are conjugate
该研究证明有限群中所有有限提升同构的子群必为共轭，并回答了Dipendra Prasad关于余集等价子群未必同构的问题，同时探讨了与 profinite 刚性、anabelian 几何等领域的联系。
A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tu…
Depth of free-by-cyclic groups
本文研究自由由循环群的叶形深度，证明其作为共轭不变量的性质，并探讨其与映射扭体结构的关系。
Let p be a prime number and G a finite group of order divisible by p. Quillen showed that the Brown poset of nonidentity p-subgroups of G is homotopy equivalent to its subposet of nonidentity elementary abelian subgroups. We show here that a similar statement holds for the fusion category of nonidentity p-subgroups of G. Other categories of p-subgroups of G are also considered.
Given two structures $\mathcal{M}$ and $\mathcal{N}$ on the same domain, we say that $\mathcal{N}$ is a reduct of $\mathcal{M}$ if all $\emptyset$-definable relations of $\mathcal{N}$ are $\emptyset$-definable in $\mathcal{M}$. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are $\aleph_0$-categorical, determining their reducts is equivalent to determining all closed supergroups $G<$ Sym$(\mathbb{N})$ of their automorphism groups. A consequence of the classific…
A pro-$p$ group $G$ is called strongly Frattini-resistant if the function $H \mapsto Φ(H)$, from the poset of all closed subgroups of $G$ into itself, is a poset embedding. Frattini-resistant pro-$p$ groups appear naturally in Galois theory. Indeed, every maximal pro-$p$ Galois group over a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is strongly Frattini-resistant. Let $G_1$ and $G_2$ be non-trivial pro-$p$ groups. We prove that $G_1 \times G_2$ is strongly Frattini-resistant if and only if one of the direct factors $G_1$ or $G_2$ is torsion-fre…
We show that a finite group which admits a faithful, smooth, orientation-preserving action on a homology 4-sphere, and in particular on the 4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by explicitly determining the various groups which can occur (up to an indetermination of index two in the case of solvable groups). As a consequence we obtain also a characterization of the finite groups which are isomorphic to subgroups of the orthogonal groups SO(5) and O(5).
The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V(ZG), the group of normalized units of the integral group ring of the finite group G, it must be isomorphic to a subgroup of G. The smallest groups known not to satisfy this property are the counterexamples to the Isomorphism Problem constructed by M. Hertweck. However the only groups known to satisfy it are cyclic groups of prime power order and elementary-abelian p-groups of rank 2. We prove the Subgroup Isomorphism Problem for C_4 x C_2. Moreover we p…
We let $φ$ be an ageometric fully irreducible outer automorphism so that its Handel-Mosher axis bundle consists of a single unique axis. We show that the centralizer $Cen(\langleφ\rangle)$ of the cyclic subgroup generated by $φ$ equals the stabilizer $\text{Stab}(Λ^+_φ)$ of the attracting lamination $Λ^+_φ$ and is isomorphic to $\mathbb Z$. We further show, via an analogous result about the commensurator, that the normalizer $N(\langleφ\rangle)$ of $\langle φ\rangle$ is isomorphic to either $\mathbb Z$ or $\mathbb Z_2 * \mathbb Z_2$.
We study a subgroup $Fr(162\times 4)$ of SU(3) of order 648 which is an extension of $D(9,1,1;2,1,1)$ and whose generators arise from anyonic systems. We show that this group is isomorphic to a semi-direct product $(\mathbb{Z}/18\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})\rtimes S_3$ with respect to conjugation and we give a presentation of the group. We show that the group $D(18,1,1;2,1,1)$ from the series $(D)$ in the existing classification for finite SU(3)-subgroups is also isomorphic to a semi-direct product $(\mathbb{Z}/18\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})\rtimes S_3$, also with respect…
Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is isomorphic to a subgroup of $G$. Furthermore, it is shown that a composition factor of a finite subgroup of $V(\mathbb{Z}G)$ is isomorphic to a subgroup of $G$.
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