arXiv subjects

Şükran Gül

Publications and source records attributed to Şükran Gül.

15 recordsLinked to original sources

arXiv2026

Sylow theory and the nilpotency class of left nilpotent skew braces

Let $X$ be a finite left nilpotent skew brace and let $π$ be a set of primes. We show that every Hall $π$-subgroup of the multiplicative group $(X,\cdot)$ is a Hall $π$-subbrace of $X$. This extends \cite[Theorem 11]{CDDFT} by removing the solvability assumption. As an application, we obtain an upper bound for the left nilpotency class of $X$ in terms of the left nilpotency classes of its Sylow $p$-subbraces for every prime $p$ dividing the order of $X$. We also show that every $p$-subbrace of $X$ is contained in some Sylow $p$-subbrace.

math.GR↗

arXiv2026

The Baer Transform for Skew Braces

We introduce and study the Baer transform for finite skew braces whose additive group has odd order and nilpotency class at most $2$. Starting from Baer's classical construction for nilpotent groups of odd order and nilpotency class $2$, we replace the additive group $(X,+)$ by an abelian group $(X,\oplus)$ on the same underlying set and prove that \[ Br(X)=(X,\oplus,\cdot) \] is a finite skew brace of abelian type. We show that the Baer transform preserves automorphisms, strong left ideals, ideals and central ideals, and we establish its compatibility with quotients. We then compare structural properties of $X$ and $Br(X)$. As applications, we show that direct product decompositions into ideals are preserved by the Baer transform, giving a criterion for indecomposability of skew braces. Finally, for finite $p$-skew braces with $p$ odd, we apply Thompson critical subgroups together with the Baer transform to embed suitable automorphism groups into automorphism groups of skew braces of abelian type.

math.GR↗

arXiv2026

The Number of Isomorphism Classes of Beauville Surfaces with Beauville $p$-Group

A Beauville surface is a rigid complex surface of general type, isogenous to a higher product by the free action of a finite group $G$, called a Beauville group. In \cite{GT}, González-Diez and Torres-Teigell find the number of isomorphism classes of Beauville surfaces for which the group $G$ is $\PSL(2,p)$ with particular types of `Beauville structures'. On the other hand, in \cite{GJT}, González-Diez, Jones and Torres-Teigell give an explicit formula for this number when the group $G$ is abelian. To the best of the author's knowledge, in the literature, the exact number of isomorphism classes of Beauville surfaces is given only for $\PSL(2,p)$ and for abelian groups. In this paper, we extend the result for Beauville surfaces with abelian $p$-group to Beauville surfaces for which the Beauville group is either a non-abelian metacyclic $p$-group or a $p$-group of nilpotency class $2$.

math.GR↗

arXiv2023

Beauville structures for quotients of generalised GGS-groups

A finite group with a Beauville structure gives rise to a certain compact complex surface called a Beauville surface. Gül and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki (GGS-)groups that act on the $p$-adic tree, for $p$ an odd prime, admit Beauville structures. We extend their result by showing that quotients of infinite periodic GGS-groups acting on $p^n$-adic trees, for $p$ any prime and $n\ge 2$, also admit Beauville structures.

math.GR↗

arXiv2022

Ramification structures for quotients of multi-EGS groups

Groups associated to surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. Gül and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGS-groups, namely multi-EGS groups, also admit ramification structures.

math.GR↗

arXiv2019

Purely (non-) strongly real Beauville $p$-groups

For every prime $p\geq5$, we give examples of Beauville $p$-groups whose Beauville structures are never strongly real. This shows that there are purely non-strongly real nilpotent Beauville groups. On the other hand, we determine infinitely many Beauville $2$-groups which are purely strongly real. This answers two questions formulated by Fairbairn in [8].

math.GR↗

arXiv2018

On the asymptotic behaviour of the number of Beauville and non-Beauville $p$-groups

We find asymptotic lower bounds for the numbers of both Beauville and non-Beauville $2$-generator finite $p$-groups of a fixed order, which turn out to coincide with the best known asymptotic lower bound for the total number of $2$-generator finite $p$-groups of the same order. This shows that both Beauville and non-Beauville groups are abundant within the family of finite $p$-groups.

math.GR↗

arXiv2018

Grigorchuk-Gupta-Sidki groups as a source for Beauville surfaces

If $G$ is a Grigorchuk-Gupta-Sidki group defined over a $p$-adic tree, where $p$ is an odd prime, we study the existence of Beauville surfaces associated to the quotients of $G$ by its level stabilizers $\st_G(n)$. We prove that if $G$ is periodic then the quotients $G/\st_G(n)$ are Beauville groups for every $n\geq 2$ if $p\geq 5$ and $n\geq 3$ if $p=3$. On the other hand, if $G$ is non-periodic, then none of the quotients $G/\st_G(n)$ are Beauville groups.

math.GR↗

arXiv2017

On ramification structures for finite nilpotent groups

We extend the characterization of abelian groups with ramification structures given by Garion and Penegini to finite nilpotent groups whose Sylow $p$-subgroups have a `nice power structure', including regular $p$-groups, powerful $p$-groups and (generalized) $p$-central $p$-groups. We also correct two errors in the result of Garion and Penegini regarding abelian $2$-groups with ramification structures and the relation between the sizes of ramification structures for an abelian group and those for its Sylow $2$-subgroup.

math.GR↗

arXiv2017

Beauville $p$-groups of wild type and groups of maximal class

Let $G$ be a Beauville finite $p$-group. If $G$ exhibits a `good behaviour' with respect to taking powers, then every lift of a Beauville structure of $G/Φ(G)$ is a Beauville structure of $G$. We say that $G$ is a Beauville $p$-group of wild type if this lifting property fails to hold. Our goal in this paper is twofold: firstly, we fully determine the Beauville groups within two large families of $p$-groups of maximal class, namely metabelian groups and groups with a maximal subgroup of class at most $2$; secondly, as a consequence of the previous result, we obtain infinitely many Beauville $p$-groups of wild type.

math.GR↗

arXiv2017

Metabelian thin Beauville $p$-groups

A non-cyclic finite $p$-group $G$ is said to be thin if every normal subgroup of $G$ lies between two consecutive terms of the lower central series and $|γ_i(G):γ_{i+1}(G)|\le p^2$ for all $i\geq 1$. In this paper, we determine Beauville structures in metabelian thin $p$-groups.

math.GR↗

arXiv2016

An infinite family of strongly real Beauville $p$-groups

We give an infinite family of non-abelian strongly real Beauville $p$-groups for every prime $p$ by considering the quotients of triangle groups, and indeed we prove that there are non-abelian strongly real Beauville $p$-groups of order $p^n$ for every $n \geq 3, 5$ or $7$ according as $p \geq 5$ or $p =3$ or $p =2$. This shows that there are strongly real Beauville $p$-groups exactly for the same orders for which there exist Beauville $p$-groups.

math.GR↗

arXiv2016

A note on strongly real Beauville $p$-groups

We give an infinite family of non-abelian strongly real Beauville $p$-groups for any odd prime $p$ by considering the lower central quotients of the free product of two cyclic groups of order $p$. This is the first known infinite family of non-abelian strongly real Beauville $p$-groups.

math.GR↗

arXiv2016

Beauville structures in $p$-central quotients

We prove a conjecture of Boston that if $p\geq 5$, all $p$-central quotients of the free group on two generators and of the free product of two cyclic groups of order $p$ are Beauville groups. In the case of the free product, we also determine Beauville structures in $p$-central quotients when $p=3$. As a consequence, we give an explicit infinite family of Beauville $3$-groups, which is different from the only one that was known up to date.

math.GR↗

arXiv2016

Beauville structures in finite p-groups

We study the existence of (unmixed) Beauville structures in finite $p$-groups, where $p$ is a prime. First of all, we extend Catanese's characterisation of abelian Beauville groups to finite $p$-groups satisfying certain conditions which are much weaker than commutativity. This result applies to all known families of $p$-groups with a good behaviour with respect to powers: regular $p$-groups, powerful $p$-groups and more generally potent $p$-groups, and (generalised) $p$-central $p$-groups. In particular, our characterisation holds for all $p$-groups of order at most $p^p$, which allows us to determine the exact number of Beauville groups of order $p^5$, for $p\ge 5$, and of order $p^6$, for $p\ge 7$. On the other hand, we determine which quotients of the Nottingham group over $\mathbb{F}_p$ are Beauville groups, for an odd prime $p$. As a consequence, we give the first explicit infinite family of Beauville $3$-groups, and we show that there are Beauville $3$-groups of order $3^n$ for every $n\ge 5$.

math.GR↗