arXiv subjects

Mariano Serrano

Publications and source records attributed to Mariano Serrano.

4 recordsLinked to original sources

arXiv2018

Zassenhaus Conjecture on torsion units holds for $\operatorname{PSL}(2,p)$ with $p$ a Fermat or Mersenne prime

H.J. Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of the form $\pm g$ with $g \in G$. Though known for some series of solvable groups, the conjecture has been proved only for thirteen non-abelian simple groups. We prove the Zassenhaus Conjecture for the groups $\operatorname{PSL}(2,p)$, where $p$ is a Fermat or Mersenne prime. This increases the list of non-abelian simple groups for which the conjecture is known by probably infinitely many, but at least by 49, groups. Our result is an easy consequence of known results and our main theorem which states that the Zassenhaus Conjecture holds for a unit in $\mathbb{Z}\operatorname{PSL}(2,q)$ of order coprime with $2q$, for some prime power $q$.

math.RA↗

arXiv2018

Zassenhaus Conjecture on torsion units holds for $\text{SL}(2,p)$ and $\text{SL}(2,p^2)$

H.J. Zassenhaus conjectured that any unit of finite order and augmentation $1$ in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of $G$. We prove the Zassenhaus Conjecture for the groups $\text{SL}(2,p)$ and $\text{SL}(2,p^2)$ with $p$ a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus Conjecture has been proved. We also prove that if $G=\text{SL}(2,p^f)$, with $f$ arbitrary and $u$ is a torsion unit of $\mathbb{Z}G$ with augmentation $1$ and order coprime with $p$ then $u$ is conjugate in $\mathbb{Q}G$ to an element of $G$. By known results, this reduces the proof of the Zassenhaus Conjecture for this groups to prove that every unit of $\mathbb{Z}G$ of order multiple of $p$ and augmentation $1$ has actually order $p$.

math.GR↗

arXiv2018

On the first Zassenhaus conjecture and direct products

In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products as well as the General Bovdi Problem (Gen-BP) which turns out to be a slightly weaker variant of (ZC1). Among others we prove that (Gen-BP) holds for Sylow tower groups, so in particular for the class of supersolvable groups. (ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group. We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G \times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

math.RA↗

arXiv2017

On the torsion units of the integral group ring of finite projective special linear groups

H. J. Zassenhaus conjectured that any unit of finite order and augmentation one in the integral group ring of a finite group $G$ is conjugate in the rational group algebra to an element of $G$. One way to verify this is showing that such unit has the same distribution of partial augmentations as an element of $G$ and the HeLP Method provides a tool to do that in some cases. In this paper we use the HeLP Method to describe the partial augmentations of a hypothetical counterexample to the conjecture for the projective special linear groups.

math.RA↗