Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is called a weakly $SΦ$-supplemented subgroup of $G$, if there exists a subgroup $T$ of $G$ such that $G =HT$ and $H \cap T \leq Φ(H) H_{sG}$, where $H_{sG}$ denotes the subgroup of $H$ generated by all subgroups of $H$ which are $S$-permutable in $G$. Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, $i=0,1,\cdots, n-1$, $S_i$ is strongly $\mathcal{F}$-closed, $i=0,1,\cdots,n$. In this paper, we investigate the structure of fusion system $\mathcal{F}_S (G)$ under the assumption that certain subgroups of $S$ are weakly $SΦ$-supplemented in $G$, and obtain several new characterizations of supersolvability of $\mathcal{F}_S (G)$.
A subgroup $A$ of a group~$G$ is said to be {\sl NS-supplemented} in $G$, if there exists a subgroup~$B$ of $G$ such that $G=AB$ and whenever $X$~is a normal subgroup of~$A$ and $p\in π(B)$, there exists a Sylow $p$-subgroup~$B_p$ of~$B$ such that $XB_p=B_pX$. In this paper, we proved the supersolubility of a group with NS-supplemented non-cyclic Sylow subgroups. The solubility of a group with NS-supplemented maximal subgroups is obtained.
A subgroup $H$ of a group $G$ is called weakly s-supplemented in $G$ if there is a subgroup $T$ such that $G=HT$ and $H\cap T\le H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are s-permutable in $G$. The influence of primary weakly s-supplemented subgroups on the structure of finite groups is investigated. An open question promoted by Skiba is studied and some known results are generalized.
Let $G$ be a finite group and $p^k$ be a prime power dividing $|G|$. A subgroup $H$ of $G$ is called to be $\mathcal{M}$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H_iK<G$ for every maximal subgroup $H_i$ of $H$. In this paper, we complete the classification of the finite groups $G$ in which all subgroups of order $p^k$ are $\mathcal{M}$-supplemented. In particular, we show that if $k\geq 2$, then $G/\mathrm{O}_{p'}(G)$ is supersolvable with a normal Sylow $p$-subgroup and a cyclic $p$-complement.
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.
A subgroup $H$ is said to be $nc$-supplemented in a group $G$ if there is a subgroup $K\leq G$ such that $HK\unlhd G$ and $H\cap K$ is contained in $H_G$, the core of $H$ in $G$. We characterize the solvability of finite groups $G$ with some subgroups of Sylow subgroups $nc$-supplemented in $G$. We also give a result on $c$-supplemented subgroups.
Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be weakly S-embedded in $G$ if there exists $K\unlhd G$ such that $HK$ is S-quasinormal in $G$ and $H\cap K\leq H_{seG}$, where $H_{seG}$ is the subgroup generated by all those subgroups of $H$ which are S-quasinormally embedded in $G$. We say that $H$ is weakly $τ$-embedded in $G$ if there exists $K\unlhd G$ such that $HK$ is S-quasinormal in $G$ and $H\cap K\leq H_{τG}$, where $H_{τG}$ is the subgroup generated by all those subgroups of $H$ which are $τ$-quasinormal in $G$. In this paper, we study the properties of the weakly S-embe…
Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups tha…
The goal of this note is to classify the weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that the Lie algebra of a connected simple algebraic group fails to be a so called 2F-module.
Let $\mathfrak{F}$ be a formation and $G$ a finite group. A subgroup $H$ of $G$ is said to be weakly $\mathfrak{F}_{s}$-quasinormal in $G$ if $G$ has an $S$-quasinormal subgroup $T$ such that $HT$ is $S$-quasinormal in $G$ and $(H\cap T)H_{G}/H_{G}\leq Z_{\mathfrak{F}}(G/H_{G})$, where $Z_{\mathfrak{F}}(G/H_{G})$ denotes the $\mathfrak{F}$-hypercenter of $G/H_{G}$. In this paper, we study the structure of finite groups by using the concept of weakly $\mathfrak{F}_{s}$-quasinormal subgroups.
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.
Let G be a finite group and σ = {σ_i, i \in I} be a partition of the set of all primes \mathbb{P}. A set \mathcal{H} of subgroups of G with 1 \in \mathcal{H} is said to be a complete Hall σ-set of G if every non-identity member of \mathcal{H} is a Hall σ_i-subgroup of G. A subgroup H of G is said to be σ-permutable if G possesses a complete Hall σ-set \mathcal{H} such that HA^x = A^xH for all A \in \mathcal{H} and all x \in G. We say that a subgroup H of G is weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H \cap T \leq H_σG. where H_σG is the subgro…
We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary representations of $G$ that are weakly contained in the left-regular representation, but not weakly equivalent to it. We discuss applications of this result and pose the problem to construct non-discrete topologically simple groups with a cocompact amenable closed subgroup but without a Gelfand pair.
A group $G$ given by a presentation $G = < \mathcal A \| \mathcal R >$ is called weakly finitely presented if every finitely generated subgroup of $G$, generated by (images of) some words in $\mathcal A^{\pm 1}$, is naturally isomorphic to the subgroup of a group $G_0 = < \mathcal A_0 \| \mathcal R_0>$, where $\mathcal A_0 \subseteq \mathcal A$, $\mathcal R_0 \subseteq \mathcal R$ are finite, generated by (images of) the same words. In the article, weakly finitely presented periodic groups which are not locally finite are constructed.
Let G and G' be absolutely almost simple algebraic groups of types B and C respectively, of rank at least 3, and defined over a number field K. We determine when G and G' have the same isomorphism or isogeny classes of maximal K-tori. This leads to the necessary and sufficient conditions for two Zariski-dense S-arithmetic subgroups of G and G' to be weakly commensurable.
The description of the subgroup structure of a non-commutative division ring is the subject of the intensive study in the theory of division rings in particular, and of the theory of skew linear groups in general. This study is still so far to be complete. In this paper, we study this problem for weakly locally finite division rings. Such division rings constitute a large class which strictly contains the class of locally finite division rings.
For a one-ended right-angled Artin group, we give an explicit description of its JSJ tree of cylinders over infinite cyclic subgroups in terms of its defining graph. This is then used to classify certain right-angled Artin groups up to quasi-isometry. In particular, we show that if two right-angled Artin groups are quasi-isometric, then their JSJ trees of cylinders are weakly equivalent. Although the converse to this is not generally true, we define quasi-isometry invariants known as stretch factors that can distinguish quasi-isometry classes of RAAGs with weakly equivalence JSJ trees of cylin…
This is the third in a series of four papers (with research announcement posted on this arXiv) that develop a decomposition theory for subgroups of $\text{Out}(F_n)$. In this paper, given an outer automorphism of $F_n$ and an attracting-repelling lamination pair, we study which lines and conjugacy classes in $F_n$ are weakly attracted to that lamination pair under forward and backward iteration respectively. For conjugacy classes, we prove Theorem F from the research annoucement, which exhibits a unique vertex group system called the "nonattracting subgroup system" having the property that the…
We construct a new family of groups that is non-contracting and weakly regular branch over the derived subgroup. This gives the first example of an infinite family of groups acting on a $d$-adic tree, with $d \geq 2$, with these properties.
We consider a generalisation of the Basilica group to all odd primes: the $p$-Basilica groups acting on the $p$-adic tree. We show that the $p$-Basilica groups have the $p$-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the $p$-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the…
A Grigorchuk-Gupta-Sidki (GGS-)group is a subgroup of the automorphism group of the $p$-adic tree for an odd prime $p$, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here we extend the result to non-torsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.
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.
We show that, in a weakly regular $p$-adic Lie group $G$, the subgroup $G_u$ spanned by the one-parameter subgroups of $G$ admits a Levi decomposition. As a consequence, there exists a regular open subgroup of $G$ which contains $G_u$.
Building on work from Sun and Kechris-Quorning, we prove that every acylindrically hyperbolic group $G$ admits a weakly mixing probability measure preserving action $G \curvearrowright (X,\mathcal{B},μ)$ which is faithful but not essentially free. In other words, $G$ admits a weakly mixing nontrivial faithful IRS. We also prove that every non-elementary hyperbolic group admits a characteristic random subgroup with the same properties.
The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly systolic complexes. As corollaries we obtain results concerning classifying spaces for the family of finite subgroups of weakly systolic groups and conjugacy classes of finite subgroups. As immediate consequences we get new results on systolic complexes and groups. The fixed point theorem is proved by using a graph-theoretical tool - dismantlability. In particular we show that 1-skeleta of weakly systolic complexes, i.e. weakly bridged graphs, are dismantlable. On the way we show numerous characte…
Given a simple Lie group $G$, we show that the lattices in $G$ are weakly uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem. Our proof however is straightforward --- considering general IRS rather than lattices allows us to apply a compactness argument. In terms of p.m.p. actions, we show that for every $ε$ there is an identity neighbourhood $U$ which intersects trivially the stabilizers of $1-ε$ of the points in every non-atomic $G$-space.
In this note, we construct torsion-free countable, amenable, weakly mixing groups, which answer a question of V. Bergelson. Some results related to verbal subgroups and crystallographic groups are also presented.
A subgroup $G$ of a product $\prod\limits_{i\in\mathbb{N}}G_i$ is \emph{rectangular} if there are subgroups $H_i$ of $G_i$ such that $G=\prod\limits_{i\in\mathbb{N}}H_i$. We say that $G$ is \emph{weakly rectangular} if there are finite subsets $F_i\subseteq \mathbb{N}$ and subgroups $H_i$ of $\bigoplus\limits_{j\in F_i} G_j$ that satisfy $G=\prod\limits_{i\in\mathbb{N}}H_i$. %We say that $G$ is a \emph{subdirect product} of the family $\{G_i\}_{i\in I}$ if $G$ is weakly rectangular and %$G\cap\bigoplus\limits_{i\in I} G_i=\bigoplus\limits_{i\in\mathbb{N}}H_i$. In this paper we discuss when a c…
To date almost all verifications of Oliver's p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3^n for n = 49 which refutes the weakly closed conjecture but satisfies Oliver's conjecture.
Let $G$ be a group. The intersection graph of cyclic subgroups of $G$, denoted by $\mathscr I_c(G)$, is a graph having all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\mathscr I_c(G)$ are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graph of cyclic subgroups is one of totally disconnected, complete, star, path, cycle. We show that for a given finite group $G$, $girth(\mathscr I_c (G)) \in \{3, \infty\}$. Moreover, we classify all finite non-cyclic abelian groups whose intersection…
Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by $\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only if the corresponding subgroups having a non-trivial intersection in $G$. In this paper, we classify the finite groups whose intersection graph of subgroups are toroidal or projective-planar. In addition, we classify the finite groups whose intersection graph of subgroups are one of bipartite, complete bipartite, tree, star graph, unicyclic, acyclic, cycle, path or total…
We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasi-regular representations is given by the double cosets …
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in $Out(F_n)$, $n\geq 3$. We prove that any simple random walk…
It is well known that if $G$ admits a f.g. subgroup $H$ with a weaklyaperiodic SFT (resp. an undecidable domino problem), then $G$itself has a weakly aperiodic SFT (resp. an undecidable domino problem).We prove that we can replace the property "$H$ is a subgroup of $G$"by "$H$ acts translation-like on $G$", provided $H$ is finitely presented.In particular:* If $G\_1$ and $G\_2$ are f.g. infinite groups, then $G\_1 \times G\_2$ has a weakly aperiodic SFT (and actually a undecidable domino problem). In particular the Grigorchuk group has an undecidable domino problem. * Every infinite f.g. $p$-g…
We initiate a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular, isometric representations on reflexive Banach spaces. Using the Ryll-Nardzewski fixed point Theorem, we prove a vanishing result for the restriction map (with respect to a subgroup) in the reduced cohomology of weakly periodic representations. Combined with the Alaoglu-Birkhoff decomposition theorem, this generalizes and complements theorems on continuous group cohomology by several authors.
In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of $\mathbb{F}_2$ which are all invariant and weakly mixing with respect to the action of $\text{Aut}(\mat…
Equationally compact subgroups of countable groups were introduced by Banaschewski. For all known cases the orbit closure of such a subgroup is a countable subset in the space of subgroups and has finite Cantor-Bendixson rank. We show that there exists a finitely generated group $Γ$ such that for any countable ordinal $α$ we have an equationally compact subgroup $H\subset Γ$ for which the Cantor-Bendixson rank of the orbit closure of $H$ equals to $α+2$. Then we give an explicite construction of continuum many equationally compact subgroups of $Γ$ such that the associated ergodic Bernoulli shi…
For a residually finite group $G$, its normal subgroups $G\supset G_1\supset G_2\cdots$ with $\cap_{n\in\mathbb N}G_n=\{e\}$ and for a growth function $γ$ we construct a unitary representation $π_γ$ of $G$. For the minimal growth, $π_γ$ is weakly equivalent to the regular representation, and for the maximal growth it is weakly equivalent to the direct sum of the quasiregular representations on the quotients $G/G_n$. In the case of intermediate growth we show two examples of different behaviour of $π_γ$.
On the torus group, on the group of p-adic integers and on the p-adic solenoid, we give a construction of an arbitrary weakly infinitely divisible probability measure using a random element with values in a product of (possibly infinitely many) subgroups of the real numbers. As a special case of our results, we have a new construction of the Haar measure on the p-adic solenoid.
We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non weak amenability results of Haagerup (1988) and Ozawa--Popa (2010). A von Neumann algebra analogue is also obtained.
In this article we prove that for any saturated fusion system, that the (unique) smallest weakly normal subsystem of it on a given strongly closed subgroup is actually normal. This has a variety of corollaries, such as the statement that the notion of a simple fusion system is independent of whether one uses weakly normal or normal subsystems. We also develop a theory of weakly normal maps, consider intersections and products of weakly normal subsystems, and the hypercentre of a fusion system.
We completely characterize connected Lie groups all of whose countable subgroups are weakly amenable. We also provide a characterization of connected semisimple Lie groups that are weakly amenable. Finally, we show that a connected Lie group is weakly amenable if the group is weakly amenable as a discrete group.
The aim of this note is to show that weak relative hyperbolicity of a group relative to a subgroup (or relative hyperbolicity in the sense of Farb) does not imply any natural analogues of some well-known algebraic properties of ordinary hyperbolic groups. Our main tools are combination theorems for weakly relatively hyperbolic groups.
We investigate invariant random subgroups in groups acting on rooted trees. Let $\mathrm{Alt}_f(T)$ be the group of finitary even automorphisms of the $d$-ary rooted tree $T$. We prove that a nontrivial ergodic IRS of $\mathrm{Alt}_f(T)$ that acts without fixed points on the boundary of $T$ contains a level stabilizer, in particular it is the random conjugate of a finite index subgroup. Applying the technique to branch groups we prove that an ergodic IRS in a finitary regular branch group contains the derived subgroup of a generalized rigid level stabilizer. We also prove that every weakly bra…
The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic groups. We have included a discussion of very recent results and conjectures on absolutely almost simple algebraic groups having the same maximal tori and finite-dimensional division algebras having the same maximal subfields.
