271 research outputs found
Torsion units in integral group rings of Janko simple groups
Using the Luthar--Passi method, we investigate the classical Zassenhaus
conjecture for the normalized unit group of integral group rings of Janko
sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel
graph of the Janko groups , and is the same as that of the
normalized unit group of their respective integral group ring.Comment: 23 pages, to appear in Math.Comp
A description of a class of finite semigroups that are near to being Malcev nilpotent
In this paper we continue the investigations on the algebraic structure of a
finite semigroup that is determined by its associated upper non-nilpotent
graph . The vertices of this graph are the elements of and
two vertices are adjacent if they generate a semigroup that is not nilpotent
(in the sense of Malcev). We introduce a class of semigroups in which the
Mal'cev nilpotent property lifts through ideal chains. We call this the class
of \B\ semigroups. The definition is such that the global information that a
semigroup is not nilpotent induces local information, i.e. some two-generated
subsemigroups are not nilpotent. It turns out that a finite monoid (in
particular, a finite group) is \B\ if and only if it is nilpotent. Our main
result is a description of \B\ finite semigroups in terms of their
associated graph . In particular, has a largest nilpotent
ideal, say , and is a 0-disjoint union of its connected components
(adjoined with a zero) with at least two elements
Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
We study relations between different notions of nilpotency in the context of
skew braces and applications to the structure of solutions to the Yang-Baxter
equation. In particular, we consider annihilator nilpotent skew braces, an
important class that turns out to be a brace-theoretic analog to the class of
nilpotent groups. In this vein, several well-known theorems in group theory are
proved in the more general setting of skew braces.Comment: 18 pages. Postprint versio
Factorizations of skew braces
We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the YangâBaxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of ItĂŽâs theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the YangâBaxter equation. Finally, we classify skew braces that contain no non-trivial proper characteristic ideals.Fil: Jespers, E.. Vrije Unviversiteit Brussel; BĂ©lgicaFil: Kubat, L.. Vrije Unviversiteit Brussel; BĂ©lgicaFil: Van Antwerpen, A.. Vrije Unviversiteit Brussel; BĂ©lgicaFil: Vendramin, Claudio Leandro. Institute of Mathematical Sciences at NYU Shanghai; China. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; Argentin
On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang--Baxter equation
Given a finite bijective non-degenerate set-theoretic solution of the
Yang--Baxter equation we characterize when its structure monoid is
Malcev nilpotent. Applying this characterization to solutions coming from
racks, we rediscover some results obtained recently by Lebed and Mortier, and
by Lebed and Vendramin on the description of finite abelian racks and quandles.
We also investigate bijective non-degenerate multipermutation (not
necessarily finite) solutions and show, for example, that this property
is equivalent to the solution associated to the structure monoid
(respectively structure group ) being a multipermuation solution and
that is solvable of derived length not exceeding the
multipermutation level of enlarged by one, generalizing results of
Gateva-Ivanova and Cameron obtained in the involutive case. Moreover, we also
prove that if is finite and is nilpotent, then the torsion part
of the group is finite, it coincides with the commutator subgroup
of the additive structure of the skew left brace and is a
trivial left brace.Comment: 35 page
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
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