Related papers: On Free Polyadic Groups

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other.…

Necessary and sufficient conditions are given for a prime Noetherian algebra K[S] of a submonoid S of a polycyclic-by-finite group G to be a maximal order. These conditions are entirely in terms of the monoid S. This extends earlier results…

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting…

Let $G$ be a free product of two groups with amalgamated subgroup, $\pi$ be either the set of all prime numbers or the one-element set \{$p$\} for some prime number $p$. Denote by $\Sigma$ the family of all cyclic subgroups of group $G$,…

Let A be an evolution algebra (possibly infinite-dimensional) equipped with a fixed natural basis B, and let E be the associated graph defined by Elduque and Labra. We describe the group of automorphisms of A that are diagonalizable with…

We define an algebraic group over a group $G$ to be a variety - that is, a subset of $G^d$ defined by equations over $G$ - endowed with a group law whose coordinates can be expressed as word maps. In the case where $G$ is a torsion-free…

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.

We study the normal subgroup <f> generated by a non trivial element f in the group G of complex plane polynomial automorphisms having Jacobian determinant 1. On one hand if f has length at most 8 relatively to the classical amalgamated…

We prove new separability results about free groups. Namely, if $H_1, \ldots , H_k$ are infinite index, finitely generated subgroups of a non-abelian free group $F$, then there exists a homomorphism onto some alternating group $f:F…

We show that if $G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous ergodic invariant random…

We show that the canonical involution on a nonabelian poly-orderable group G extends to the Hughes-free division ring of fractions D of the group algebra k[G] of G over a field k and that, with respect to this involution, D contains a pair…

We study in detail the profinite group G arising as geometric \'etale iterated monodromy group of an arbitrary quadratic polynomial over a field of characteristic different from two. This is a self-similar closed subgroup of the group of…

Given an action of a group G on a topological space X, we establish a necessary and sufficient condition for the existence of a free subgroup F of rank 2 of G acting properly discontinuously on at least one nonempty, open, F-invariant…

Let $G$ be a connected semisimple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers and $B$ be a Borel subgroup of $G.$ Let $F$ be an irreducible projective $B$-variety. Then consider the variety…

Let $\Phi:F\rightarrow F$ be an automorphism of the finite-rank free group $F$. Suppose that $G=F\rtimes_\Phi\mathbb Z$ is word-hyperbolic. Then $G$ acts freely and cocompactly on a CAT(0) cube complex.

We give a counterexample to a conjecture by Miasnikov, Ventura and Weil, stating that an extension of free groups is algebraic if and only if the corresponding morphism of their core graphs is onto, for every basis of the ambient group. In…

We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group…

Given a finitely generated subgroup $H$ of a free group $F$, we present an algorithm which computes $g_1,\ldots,g_m\in F$, such that the set of elements $g\in F$, for which there exists a non-trivial $H$-equation having $g$ as a solution,…

We extend the class of abelian groups for which a conjecture of Asai and Yoshida on the number of crossed homomorphisms holds. We also prove a general result which connects certain problems concerning divisibility in groups to the…

Let G be a finite abelian group and F a field such that char(F) does not divide |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I and J of FG are G-equivalent if there exists an…