Related papers: The commutator subgroup of the braid group is gene…
A group in which every element commutes with its endomorphic images is called an $E$-group. Our main result is that all 3-generator $E$-groups are abelian. It follows that the minimal number of generators of a finitely generated non-abelian…
Let S=Sym(\Omega) be the group of all permutations of an infinite set \Omega. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, respectively as a monoid, then there exists a…
We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type A.…
In this paper, we introduce a novel approach for generating random elements of a finite group given a set of generators of that. Our method draws upon combinatorial group theory and automata theory to achieve this objective. Furthermore, we…
We consider braids on $m+n$ strands, such that the first $m$ strands are trivially fixed. We denote the set of all such braids by $B_{m,n}$. Via concatenation $B_{m,n}$ acquires a group structure. The objective of this paper is to find a…
Let G be a group generated by $r$ elements $g_1,g_2,..., g_r.$ Among the reduced words in $g_1,g_2,..., g_r$ of length $n$ some, say $\gamma_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that…
We define pseudo-Garside groups and prove a theorem about them parallel to Garside's result on the word problem for the usual braid groups. The main novelty is that the set of simple elements can be infinite. We introduce a group B=B(Z^n)…
A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that…
After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete. The most recently updated list can be found in P.O. Ludl, J. Phys. A: Math. Theor. 44 255204 (2011), where the structure of the series (C)…
We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable…
This chapter provides a comprehensive survey of foundational results and recent advances concerning minimal generating sets for the mapping class group of a nonorientable surface, $\mathrm{Mod}(N_{g})$, and its index-two twist subgroup,…
A $2-(n,4,\lambda)$ design $(\Omega, \mathcal{B})$ is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym$(\Omega)$ called a "Conway groupoid". The…
Let $G$ be the first Grigorchuk group. We show that the commutator width of $G$ is $2$: every element $g\in [G,G]$ is a product of two commutators, and also of six conjugates of $a$. Furthermore, we show that every finitely generated…
Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p-subgroups. Quasi-split simply-connected semisimple groups can be described in the combinatorial terms of…
We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking…
In his initial paper on braids E.Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin's presentation for various generalizations of braids.
We give an explicit geometric argument that Artin's braid group $B_n$ is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical…
Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author…
For the elements of a numerical semigroup which are larger than the Frobenius number, we introduce the definition of, seed, by broadening the notion of generator. This new concept allows us to explore the semigroup tree in an alternative…
We study groups generated by sets of pattern avoiding permutations. In the first part of the paper we prove some general results concerning the structure of such groups. In the second part we carry out a case-by-case analysis of groups…