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Ángel del Río

Publications and source records attributed to Ángel del Río.

At least 19 recordsLinked to original sources
arXiv2026

The Gruenberg-Kegel graph of finite solvable groups that are character-quadratic or semi-rational

A finite group $G$ is said to be semi-rational if the set of generators of each cyclic subgroup of $G$ is contained in at most two $G$-conjugacy classes. This is equivalent to the following condition: for every column of the character table of $G$, the values appearing in the column are contained in a quadratic extension of the field of rational numbers (possibly a different one for each column). When the analogous condition holds for the rows, that is, when the field of values of every irreducible character is contained in a quadratic extension of the rationals, we say that the group is character-quadratic (these groups are often called quadratic rational in the literature). We obtain several results concerning the structure of the Gruenberg-Kegel graph of a finite solvable group that is either character-quadratic or semi-rational. More precisely, we first provide a complete classification of such graphs in the disconnected case. Also, we prove that if the graph has at most three vertices and the group is nontrivial, then it belongs to an explicit list of $20$ graphs (in the semi-rational case, this result is proved under the additional assumption that the order of the group is not divisible by $17$), and all of them are realizable except perhaps one. Finally, we show that if the graph has four vertices, then it must have at least four edges.

math.GR↗
arXiv2026

Gruenberg-Kegel graphs: cut groups, rational groups and the Prime Graph Question

The Gruenberg-Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices $p$, $q$ are joined by an edge whenever the group has an element of order $pq$. It reflects interesting properties of the group. A group is said to be cut if the central units of its integral group ring are trivial. This is a rich family of groups, which contains the well studied class of rational groups, and has received attention recently. In the first part of this paper we give a complete classification of the Gruenberg-Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg-Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg-Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg-Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and $2$-Frobenius groups for the classes of cut groups and rational groups, respectively. The Prime Graph Question asks whether the Gruenberg-Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. The recent appearance of a counter-example for the First Zassenhaus Conjecture on the torsion units of integral group rings has highlighted the relevance of this question. We answer the Prime Graph Question for integral group rings for finite rational groups and most finite cut groups.

math.GR↗
arXiv2026

Gruenberg-Kegel graphs with four edges of finite solvable cut groups

We prove that there is a unique graph with four edges which is the Gruenberg-Kegel graph of a solvable cut group. This contributes to the classification of the Gruenberg-Kegel graphs of solvable cut groups initiated in [A. Bächle, A. Kiefer, S. Maheshwary, and A. del Río. "Gruenberg-Kegel graphs: Cut groups, rational groups and the prime graph question" Forum Mathematicum, vol. 35, no. 2, 2023, pp. 409-429. https://doi.org/10.1515/forum-2022-0086].

math.GR↗
arXiv2025

Groupoid graded rings and their categories of graded modules

Let $G$ be a groupoid acting on a set $X$ and let $R$ be a $G$-graded ring with graded local units. We study the main properties of the category $gr-(R,G,X)$ of $X$-graded $R$-modules and adjoint functors between categories of this kind. We characterize the latter in terms of tensor-like and hom-like functors. As an application we obtain a characterization of equivalences between such categories in the spirit of Morita theory. Then we introduce restriction and induction functors between categories of type $gr-(R,G,X)$ and show that many functors between such categories can be realized naturally as a restriction or induction functor. This includes the forgetful functor, the functor associating a $G$ graded $R$-module with the $H$-graded module formed by the sum of the homogeneous components of degree in subgroupoid $H$, and the one associating it with the collapsing of homogeneous components in cosets modulo $H$ for $H$ a wide subgrupoid. We characterize when a restriction or induction functor is an equivalence of categories. Finally, we prove that $gr-(R,G,X)$ is always equivalent to the category of modules over a ring with local units and, generalizing a result of Menini and Năstăsescu, we characterize when $gr-(R,G,X)$ is equivalent to the category of modules over a unital ring.

math.RA↗
arXiv2025

Finite groups in integral group rings

Notes used for a course held in 2016 in the School of Advances in Group Theory and Applications, for some lectures given in 2018 for the students of the Master in Mathematics of the Vrije Universiteit Brussels, a course for master and Ph.D. students at the Universidade de São Paulo and at the conference Group algebras, representations and computations. We revise some problems on the study of finite subgroups of the group of units of integral group rings of finite groups and some techniques to attack them.

math.RA↗
arXiv2025

Uniformly semi-rational groups

We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which we call uniformly semi-rational groups. Moreover, we associate to every finite group two invariants, called rationality and semi-rationality of the group. They measure respectively how far a group is from being rational and how much uniformly rational it is. We determine the possible values that these invariants may take for finite nilpotent groups. We also classify the fields that can occur as the field generated by the character values of a finite nilpotent group.

math.GR↗
arXiv2025

Aritmethic lattices of $\SO(1,n)$ and units of group rings

We establish that standard arithmetic subgroups of a special orthogonal group ${\rm SO}(1,n)$ are conjugacy separable. As an application we deduce this property for unit groups of certain integer group rings. We also prove that finite quotients of group of units of any of these group rings determines the original group ring.

math.GR↗
arXiv2024

On commutative tensor factors of group algebras

We prove that any tensor product factorization with a commutative factor of a modular group algebra over a prime field comes from a direct product decomposition of the group basis. This extends previous work by Carlson and Kovács for the commutative case and answers a question of them in some cases.

math.RT↗
arXiv2024

On the Modular Isomorphism Problem for 2-generated groups with cyclic derived subgroup

We continue the analysis of the Modular Isomorphism Problem for $2$-generated $p$-groups with cyclic derived subgroup, $p>2$, started in [D. García-Lucas, Á. del Río, and M. Stanojkovski. On group invariants determined by modular group algebras: even versus odd characteristic. Algebr. Represent. Theory. https://doi.org/10.1007/s10468-022-10182-x, 2022]. We show that if $G$ belongs to this class of groups, then the isomorphism type of the quotients $G/(G')^{p^3}$ and $G/γ_3(G)^p$ are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification \cite{OsnelDiegoAngel}. We also show that for groups in this class of order at most $p^{11}$, the Modular Isomorphism Problem has positive answer. Finally, we describe some families of groups of order $p^{12}$ whose group algebras over the field with $p$ elements cannot be distinguished with the techniques available to us.

math.GR↗
arXiv2024

The Gruenberg-Kegel graph of finite solvable rational groups

A finite group $G$ is said to be rational if every character of $G$ is rational-valued. The Gruenberg-Kegel graph of a finite group $G$ is the undirected graph whose vertices are the primes dividing the order of $G$ and the edges join different primes $p$ and $q$ whenever $G$ contains an element of order $pq$. In this paper, we complete the classification of the Gruenberg-Kegel graphs of finite solvable rational groups initiated in \cite{BKMdR}.

math.GR↗
arXiv2024

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism from FG to Fn which maps G to the standard basis of Fn. Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space Fn, which does not assume an a priori group algebra structure on Fn. As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.

cs.IT↗
arXiv2023

A classification of metacyclic groups by group invariants

We obtain a new classification of the finite metacyclic group in terms of group invariants. We present an algorithm to compute these invariants, and hence to decide if two given finite metacyclic groups are isomorphic, and another algorithm which computes all the metacyclic groups of a given order. A GAP implementation of these algorithms is given.

math.GR↗
arXiv2023

A classification of the finite two-generated cyclic-by-abelian groups of prime power order

We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime-power order. For that we associate to each such group $G$ a list $\inv(G)$ of numerical group invariants which determines the isomorphism type of $G$. Then we describe the set formed by all the possible values of $\inv(G)$. This allows computer implementations for constructing all the finite-two generated cyclic-by-abelian groups of a given prime-power order, computing the invariants of such a group, and to decide whether two such groups are isomorphic.

math.GR↗
arXiv2022

On group invariants determined by modular group algebras: even versus odd characteristic

Let $p$ be a an odd prime and let $G$ be a finite $p$-group with cyclic commutator subgroup $G'$. We prove that the exponent and the abelianization of the centralizer of $G'$ in $G$ are determined by the group algebra of $G$ over any field of characteristic $p$. If, additionally, $G$ is $2$-generated then almost all the numerical invariants determining $G$ up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of $G'$ is determined. These claims are known to be false for $p=2$.

math.GR↗