We investigate one question regarding bicrossed products of finite groups which we believe has the potential of being approachable for other classes of algebraic objects (algebras, Hopf algebras). The problem is to classify the groups that can be written as bicrossed products between groups of fixed isomorphism types. The groups obtained as bicrossed products of two finite cyclic groups, one being of prime order, are described.
We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, α, β)$ is deformed using a combinatorial datum $(σ, v, r)$ consisting of an automorphism $σ$ of $H$, a permutation $v$ of the set $G$ and a transition map $r: G\to H$ in order to obtain a new matched pair $\bigl(H, (G,*), α', β' \bigl)$ such that there exist an $σ$-invariant isomorphism of groups $H {}_α \bowtie_β G \cong H {}_{α'} \bowtie_{β'} (G,*)$. Moreover, if we fix the group $H$ and the automorphism $σ\in \Aut(H)$ then any $σ$-invariant isomorphism …
Let $H$ be a group and $E$ a set such that $H \subseteq E$. We shall describe and classify up to an isomorphism of groups that stabilizes $H$ the set of all group structures that can be defined on $E$ such that $H$ is a subgroup of $E$. A general product, which we call the unified product, is constructed such that both the crossed product and the bicrossed product of two groups are special cases of it. It is associated to $H$ and to a system $\bigl((S, 1_S,\ast), \triangleleft, \, \triangleright, \, f \bigl)$ called a group extending structure and we denote it by $H \ltimes S$. There exists a …
We use Bieri-Strebel invariants to determine when a normal fibre product in the product of two finitely presented groups is finitely presented. We give conditions that imply and in some cases characterize the existence of such finitely presented fibre products.
There has been substantial investigation in recent years of subdirect products of limit groups and their finite presentability and homological finiteness properties. To contrast the results obtained for limit groups, Baumslag, Bridson, Holt and Miller investigated subdirect products (fibre products) of finitely presented metabelian groups. They showed that, in contrast to the case for limit groups, such subdirect products could have diverse behaviour with respect to finite presentability. We show that, in a sense that can be made precise, `most' subdirect products of a finite set of finitely p…
The purpose of this paper is two fold. First we introduce the box-tensor product of two groups as a generalization of the nonabelian tensor product of groups. We extend various results for nonabelian tensor products to the box-tensor product such as the finiteness of the product when each factor is finite. This would give yet another proof of Ellis's theorem on the finiteness of the nonabelian tensor product of groups when each factor is finite. Secondly, using the methods developed in proving the finiteness of the box-tensor product, we prove the finiteness of Inassaridze's tensor product und…
Let $\mathcal{S}$ be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in $\mathcal{S}$ is topologically finitely generated, provided that the actions of the groups in $\mathcal{S}$ are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in $\mathcal{S}$ are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterat…
A notion of \emph{graph-wreath product} is introduced. We obtain sufficient conditions for these products to satisfy the topologically inspired finiteness condition type $\operatorname{F}_n$. Under various additional assumptions we show that these conditions are necessary. Our results generalise results of Cornulier about wreath products in case $n=2$. Graph-wreath products include classical permutational wreath products and semidirect products of right-angled Artin groups by groups of automorphisms amongst others.
Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A \ast B$. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to $A$ and $B$. This proves a conjecture of Tamburini and Wilson.
We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finite…
In this paper we study the residual solvability of the generalized free product of finitely generated nilpotent groups. We show that these kinds of structures are often residually solvable.
We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of Lie type of bounded rank. Some of our arguments apply recent advances in the theory of growth in finite simple groups of Lie type, and provide a variety of new product decompositions of these groups.
We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality.
Let G be a finite p-group, where p is a prime number, and $χ$ and $ψ$ be faithful complex irreducible characters of G. We study the relation between the number $η(χ,ψ)$ of distinct irreducible constituents of the product $χψ$ and p.
Let $G$ be a finite $p$-group and $χ,ψ$ be irreducible characters of $G$. We study the character $χψ$ when $χψ$ has at most $p-1$ distinct irreducible constituents.
We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C_1,...,C_n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C_1,...,C_n and the minimal number of generators of G is n.
All possible products of all elements of an odd order finite group are considered. A set of all such products is called as a K-set. A hypothesis of K-set coincidence of any group of an odd order with its commutant is proposed and the hypothesis validity for groups with a commutant of a simple structure is shown.
We partly generalize the estimate for the rank of intersection of subgroups in free products of groups, proved earlier by S.V.Ivanov and W.Dicks, to the case of free amalgamated products of groups with normal finite amalgamated subgroup. We also prove that the obtained estimate is sharp and cannot be further improved when the amalgamated product contains an involution.
We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.
We classify those sequences $\langle S_{n} \mid n \in \mathbb{N} \rangle$ of finite simple nonabelian groups such that the full product $\prod_{n} S_{n}$ has property (FA).
We say that a finitely generated group $G$ has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. A quasi-tree is a connected graph with path metric quasi-isometric to a tree, and product spaces are equipped with the $\ell^1$-metric. As an application of the projection complex techniques, we prove that residually finite hyperbolic groups and mapping class groups have (QT).
Following partially a suggestion by Pyber, we prove that the diameter of a product of non-abelian finite simple groups is bounded linearly by the maximum diameter of its factors. For completeness, we include the case of abelian factors and give explicit constants in all bounds.
We classify all finite groups G such that the product of any two non-inverse conjugacy classes of G is always a conjugacy class of G. We also classify all finite groups G for which the product of any two G-conjugacy classes which are not inverse modulo the center of G is again a conjugacy class of G.
We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagon…
We show that the wreath product $G \wr \mathbb{Z}^n$ of any finitely generated group $G$ with $\mathbb{Z}^n$ has finite palindromic width. We also show that $C \wr A$ has finite palindromic width if $C$ has finite commutator width and $A$ is a finitely generated infinite abelian group. Further we prove that if $H$ is a non-abelian group with finite palindromic width and $G$ any finitely generated group, then every element of the subgroup $G' \wr H$ can be expressed as a product of uniformly boundedly many palindromes. From this we obtain that $P \wr H$ has finite palindromic width if $P$ is a …
In the context of group-theoretic fast matrix multiplication the TPP capacity is used to bound the exponent $ω$ of matrix multiplication. We prove a new and sharper upper bound for the TPP subgroup capacity of a finite group
We examine the question of which finitely generated groups act properly on a finite product of simplicial trees, considering both arbitrary trees and where all trees are locally finite. In the second case we present evidence in favour of hyperbolic surface groups having such an action. However we also present evidence that many RAAGs do not admit such an action and we give an example of a virtually special group which does not act properly preserving factors on any finite product of locally finite trees, even though it does so on a product of three trees without the local finiteness condition.
It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).
We construct torsion-free hyperbolic groups without unique product whose subgroups up to some given finite index are themselves non-unique product groups. This is achieved by generalising a construction of Comerford to graphical small cancellation presentations, showing that for every subgroup $H$ of a graphical small cancellation group there exists a free group $F$ such that $H*F$ admits a graphical small cancellation presentation.
We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to …
Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S'$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free product of $\Z$ with a $\Z^n$ for each $n$ occurring in $S'$.
We prove existence of the large deviation principle, with a proper convex rate function, for the distribution of the renormalized distance from the origin of a random walk on a free product of finitely generated groups. As a consequence, we derive the same principle for nearest-neighbour random walks on regular trees.
Let R be a class of groups closed under taking semidirect products with finite kernel and fully residually R-groups. We prove that R contains all R-by-{finitely generated residually finite} groups. It follows that a semidirect product of a finitely generated residually finite group with a surjunctive group is surjunctive. This remained unknown even for direct products of a surjunctive group with the integers Z.
