arXiv subjects

Mauricio Caicedo

Publications and source records attributed to Mauricio Caicedo.

7 recordsLinked to original sources

arXiv2021

Global and local properties of finite groups with only finitely many central units in their integral group ring

The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same, in particular the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.

math.GR↗

arXiv2021

Orders of units in integral group rings and blocks of defect $1$

We show that if the Sylow $p$-subgroup of a finite group $G$ is of order $p$, then the normalized unit group of the integral group ring of $G$ contains a normalized unit of order $pq$ if and only if $G$ contains an element of order $pq$, where $q$ is any prime. We use this result to answer the Prime Graph Question for most sporadic simple groups and some simple groups of Lie type, including a new infinite series of such groups. Our methods are based on the understanding of blocks of cyclic defect and Young tableaux combinatorics.

math.GR↗

arXiv2019

On the Zassenhaus Conjecture for certain cyclic-by-nilpotent groups

Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently for metabelian groups, by Eisele and Margolis. However it is known to be true for many classes of solvable groups, as for example nilpotent groups, cyclic-by-abelian groups and groups having a cyclic Sylow subgroup with abelian complement. On the other hand, it is not known whether the conjecture holds for supersolvable groups. This paper is a contribution to this question. More precisely, we study the conjecture for the class of cyclic-by-nilpotent groups with special attention to the class of cyclic-by-Hamiltonian groups. We prove the conjecture for cyclic-by-$p$-groups and for a large class of cyclic-by-Hamiltonian groups.

math.GR↗

arXiv2017

On the Prime Graph Question for Almost Simple Groups with an Alternating Socle

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and $q$ are primes greater than $\frac{n}{3}$. We combine this with some explicit computations to verify the Prime Graph Question for all almost simple groups with socle $A_n$ if $n \leq 17$.

math.GR↗

arXiv2015

A classification of exceptional components in group algebras over abelian number fields

When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ the ring of integers of an abelian number field $F$ and a finite group $G$) certain components in the Wedderburn decomposition of $FG$ cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 are division rings, type 2 are $2 \times 2$-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (w.r.t. quotients) having those division rings in their Wedderburn decomposition over $F$. We also classify the exceptional components of type 2 appearing in group algebras of a finite group over number fields $F$ by describing all 58 finite groups $G$ having a faithful exceptional Wedderburn component of this type in $FG$.

math.RT↗

arXiv2013

On the Congruence Subgroup Problem for integral group rings

Let $G$ be a finite group, $\Z G$ the integral group ring of $G$ and $\U(\Z G)$ the group of units of $\Z G$. The Congruence Subgroup Problem for $\U(\Z G)$ is the problem of deciding if every subgroup of finite index of $\U(\Z G)$ contains a congruence subgroup, i.e. the kernel of the natural homomorphism $\U(\Z G) \rightarrow \U(\Z G/m\Z G)$ for some positive integer $m$. The congruence kernel of $\U(\Z G)$ is the kernel of the natural map from the completion of $\U(\Z G)$ with respect to the profinite topology to the completion with respect to the topology defined by the congruence subgroups. The Congruence Subgroup Problem has a positive solution if and only if the congruence kernel is trivial. We obtain an approximation to the problem of classifying the finite groups for which the congruence kernel of $\U(\Z G)$ is finite. More precisely, we obtain a list $L$ formed by three families of finite groups and 19 additional groups such that if the congruence kernel of $\U(\Z G)$ is infinite then $G$ has an epimorphic image isomorphic to one of the groups of $L$. About the converse of this statement we at least know that if one of the 19 additional groups in $L$ is isomorphic to an epimorphic image of $G$ then the congruence kernel of $\U(\Z G)$ is infinite. However, to decide for the finiteness of the congruence kernel in case $G$ has an epimorphic image isomorphic to one of the groups in the three families of $L$ one needs to know if the congruence kernel of the group of units of an order in some specific division algebras is finite and this seems a difficult problem.

math.GR↗

arXiv2012

Zassenhaus conjecture for cyclic-by-abelian groups

Zassenhaus Conjecture for torsion units states that every augmentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cyclic-by-abelian groups.

math.RT↗