The family of semiabelian p-groups is the minimal family that contains {1} and is closed under quotients and semidirect products with finite abelian p-groups. Kisilevsky and Sonn have solved the minimal ramification problem for a certain subfamily G_p of the family of semiabelian p-groups. We show that G_p is in fact the entire family of semiabelian p-groups and by this complete their solution to all semiabelian p-groups.
Groups associated to surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. Gül and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGS-groups, namely multi-EGS groups, also admit ramification structures.
Using the computer algebra program GAP, we show that all crystallographic groups in dimensions at most 4 are distinguished from each other by their sets of finite quotients.
These notes are based on the mini-course "On the Graham Higman group", given at the Erwin Schrödinger Institute in Vienna, January 20, 22, 27 and 29, 2016, as a part of the Measured Group Theory program. The main purpose is to describe p-quotients of the Higman group $H(k)$ for $p|(k-1)$. (One may check that the condition $p|(k-1)$ is necessary for the existence of such quotients.)
We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can be even not recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem while still not…
This paper describes some generalizations of the results presented in the book "Geometry of defining Relations in Groups" , of A.Yu.Ol'shanskii to the case of non-cyclic torsion-free hyperbolic groups. In particular, it is proved that for every non-cyclic torsion-free hyperbolic group, there exists a non-Abelian torsion-free quotient group in which all proper subgroups are cyclic, and the intersection of any two of them is not trivial.
2-blocks with abelian defect groups and inertial quotient of prime order
School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）；Faculty of Mathematics and Statistics, Hubei University（湖北大学数学与统计学院）
Groups associated to surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. In this paper, we prove that infinitely many quotients of the Grigorchuk groups admit ramification structures. This gives the first explicit infinite family of 3-generated finite 2-groups with ramification structures.
For the class of solvable groups of homeomorphisms of the line preserving orientation and containing a freely acting element, we establish the metabelianity of the quotient group $G/H_G$, where the elements of the normal subgroup $H_G$ are stabilizers of the minimal set. This fact is an important element in the classification theorem, used, in particular, in the study of the Thompson's group $F$.
In this expanded account of a talk given at the Oberwolfach Arbeitsgemeinschaft "Totally Disconnected Groups", October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is a paper, entitled "Generators and commutators in finite groups; abstract quotients of compact groups", that was published by Nikolov and Segal in 2012. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminat…
It is well-known that random groups with at least as many relators as generators have vanishing first betti number with high probability. In this paper we extend this question and study maps from few-relators random groups to infinite virtually abelian groups.
Let $H$ be a closed subgroup of a regular abelian paratopological group $G$. The group reflexion $G^\flat$ of $G$ is the group $G$ endowed with the strongest group topology, weaker that the original topology of $G$. We show that the quotient $G/H$ is Hausdorff (and regular) if $H$ is closed (and locally compact) in $G^\flat$. On the other hand, we construct an example of a regular abelian paratopological group $G$ containing a closed discrete subgroup $H$ such that the quotient $G/H$ is Hausdorff but not regular.
We offer in this note a self-contained proof of the fact that a finitely generated group is not virtually nilpotent if and only if it has a quotient with the infinite conjugacy class (ICC) propoerty. This proof is a modern presentation of the original proof, by McLain (1956) and Duguid and McLain (1956).
The theory of rough sets was firstly introduced by Pawlak (see \cite{p}). Many Mathematician has been studied the relations between rough sets and algebraic systems such as groups, rings and modules. In this paper we will introduce the lower and upper approximations in a quotient group. We will discuss several properties of the lower and upper approximations. Moreover under some additional assumptions we are able to show that the lower approximation is a normal subgroup of the quotient group but this property fails for the upper approximation. At the end we will develop several homomorphisms b…
In this paper we introduce and study the lattice of normal subgroups of a group $G$ that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of $G$ (see \cite{5}). A precise description of this lattice is given for some particular classes of finite groups.
In this article, we show that a group $G$ is the union of two proper subsemigroups if and only if $G$ has a nontrivial left-orderable quotient. Furthermore, if $G$ is the union of two proper semigroups, then there exists a minimum normal subgroup $N\unlhd G$ for which $G/N$ is left-orderable and nontrivial.
We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_Γ$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_Γ)$ virtually surjects onto some infinite Coxeter group. In particular, the group ${\rm Aut}(W_Γ)$ is virtually indicable and therefore does not have Kazhdan's property (T).
We present a natural extension of the process of taking a group quotient to arbitrary subgroups. We first review basic concepts from group theory. This will allow us to see the relationship between our new, more general quotient operation and the standard group quotient. In particular, we will find that a naive attempt to perform the quotient process with a nonnormal subgroup actually leads to a well-defined operation which can be easily characterized with existing terminology.
In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the $i$-th $\ell^2$-Betti number is not a profinite invariant for all $i \ge 2$.
Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.
We give an new, elementary proof of the result that the smallest non-cyclic quotients of automorphism group of free group is the linear group over the field of two elements, and moreover all minimal quotients are obtained by the standard projection composed with an automorphism of the image. This result, originally due to Baumeister-Kielak-Pierro, proves a conjecture of Mecchia-Zimmermann.
We prove that the smallest non-trivial quotients of the commutator subgroups of the braid groups are the alternating groups, proving a conjecture of Chudnovsky-Kordek-Li-Partin. Furthermore, we show that any minimal quotient map is the standard projection, composed with an automorphism of the alternating group.
We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group.
