arXiv subjects

Francesco Fumagalli

Publications and source records attributed to Francesco Fumagalli.

17 recordsLinked to original sources

arXiv2025

Enhanced power graphs of finite groups with cograph structure

The enhanced power graph, $\mathcal{E}(G)$, of a group $G$ has vertex set $G$ and two elements are adjacent if they generate a cyclic subgroup. In the case of finite groups, we identify some striking and unexpected properties of these graphs, as well as links between properties of $\mathcal{E}(G)$ and properties of the group $G$. We prove that if $\mathcal{E}(G)$ is a cograph then it is also a chordal graph. Making use of properties of simplicial vertices, we characterise the finite groups $G$ whose enhanced power graph is diamond-free or a block graph. We also characterise the finite groups having enhanced power graph a cograph or a quasi-threshold graph, and those with $C_4$-free enhanced power graph. We use these characterisations to classify the finite nonabelian simple groups whose enhanced power graph is a cograph and give information on the finite simple groups whose enhanced power graph is $C_4$-free. Some open problems are posed.

math.GR↗

arXiv2025

Graphene nanowindows as a basis for creating mechanically robust nanohydroxyapatite bone lamellar scaffolds

The role of graphene nanowindows in the nanohydroxyapatite bone scaffold preparation is important for the preparation of mechanically robust scaffolds and implementation in bone recovery. Here, we report a graphene-nanohydroxyapatite (G-nHAP) scaffold synthesized by the hydrothermal method, along with its formation mechanism and promising in vivo applications. The G-nHAP scaffold exhibits excellent mechanical strength comparable to that of cancellous bone. The nHAP was grown on 2D layers of graphene with nanowindows. The role of nanowindows was to attract electrostatically Ca2+, PO43+, and OH- precursors of nHAP, forming a layered structure of G-nHAP, in which nHAP nanorods of 23.8 nm in diameter and 84.7 nm in length were placed between graphene layers, as evidenced with molecular dynamic simulations. In vivo study showed mature and mineralized bone and osteoid after six weeks. This demonstrates the role of graphene nanowindows in the formation of nHAP scaffolds that are promising for future implementation in bone tissue regeneration.

physics.chem-ph↗

arXiv2025

The elusive chase for the first RR Lyr star in a binary system: the case of KIC 2831097

The lack of RR Lyr stars in binary systems is an atypical fact when we compared it to other classes of variables. Therefore, it has become a challenge for observers to detect an RR Lyr variable in a binary system. The RR Lyr variable KIC 2831097 was one of the most promising candidates. The phases of maximum brightness in the Kepler photometry showed a regular variation superimposed on a parabolic trend. These variations in the times of maximum brightness (Tmax ) were interpreted as a possible light-time travel effect (LTTE) in a wide binary and a fast evolutionary change in the period. We planned two spectroscopic runs with the FIES instrument mounted at the NOT to test the hypothesis of binarity. The observations were programmed at the predicted quadratures of the orbit. The GEOS collaboration complemented the spectroscopic survey by a photometric one. We also analysed Gaia time series and intensive TESS photometry. The RV curves obtained at the quadratures show the same mean RV (-203 km/s), which rules the possibility of an LTTE out. KIC 2831097 is a single high-velocity metal-poor RRc star belonging to the Galactic halo. We revisited Kepler photometry and detected a weak Blazhko effect consisting of an oscillation of only 1.1% of the period in about 50 d. We also analysed the TESS photometry of Kepler-1601, whose photometry is contaminated by KIC 2831097. In total, we collected 3624 times of maximum brightness. Linear ephemerides cannot fit the whole dataset, but only parts of them. The period shows a tendency to decrease in value, as if it were an evolutionary effect, but not at a constant rate.

astro-ph.SR↗

arXiv2024

On the Gowers trick for classical simple groups

If $A$, $B$, $C$ are subsets in a finite simple group of Lie type $G$ at least two of which are normal with $|A||B||C|$ relatively large, then we establish a stronger conclusion than $ABC = G$. This is related to a theorem of Gowers and is a generalization of a theorem of Larsen, Shalev, Tiep and the second author and Pyber.

math.GR↗

arXiv2023

EEG Synthetic Data Generation Using Probabilistic Diffusion Models

Electroencephalography (EEG) plays a significant role in the Brain Computer Interface (BCI) domain, due to its non-invasive nature, low cost, and ease of use, making it a highly desirable option for widespread adoption by the general public. This technology is commonly used in conjunction with deep learning techniques, the success of which is largely dependent on the quality and quantity of data used for training. To address the challenge of obtaining sufficient EEG data from individual participants while minimizing user effort and maintaining accuracy, this study proposes an advanced methodology for data augmentation: generating synthetic EEG data using denoising diffusion probabilistic models. The synthetic data are generated from electrode-frequency distribution maps (EFDMs) of emotionally labeled EEG recordings. To assess the validity of the synthetic data generated, both a qualitative and a quantitative comparison with real EEG data were successfully conducted. This study opens up the possibility for an open\textendash source accessible and versatile toolbox that can process and generate data in both time and frequency dimensions, regardless of the number of channels involved. Finally, the proposed methodology has potential implications for the broader field of neuroscience research by enabling the creation of large, publicly available synthetic EEG datasets without privacy concerns.

eess.SP↗

arXiv2022

An upper bound for the nonsolvable length of a finite group in terms of its shortest law

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in $G$, is called the \emph{nonsolvable length} $λ(G)$ of $G$. In the present paper, we prove a theorem about permutation representations of groups of fixed nonsolvable length. As a consequence, we show that in a finite group of nonsolvable length at least $n$, no non-trivial word of length at most $n$ (in any number of variables) can be a law. This result is then used to give a bound on $λ(G)$ in terms of the length of the shortest law of $G$, thus confirming a conjecture of Larsen. Moreover our Theorem C can be used to give a positive answer, in the case $p=2$, to a problem raised by Khukhro and Shumyatsky, concerning the non-$p$-solvable length of finite groups.

math.GR↗

arXiv2022

On some questions related to integrable groups

A group $G$ is integrable if it is isomorphic to the derived subgroup of a group $H$; that is, if $H'\simeq G$, and in this case $H$ is an integral of $G$. If $G$ is a subgroup of $U$, we say that $G$ is integrable within $U$ if $G=H'$ for some $H\leq U$. In this work we focus on two problems posed in [1]. We classify the almost-simple finite groups $G$ that are integrable, which we show to be equivalent to those integrable within $\mathrm{Aut}(S)$, where $S$ is the socle of $G$. We then classify all $2$-homogeneous subgroups of the finite symmetric group $S_n$ that are integrable within $S_n$.

math.GR↗

arXiv2022

On the maximal number of elements pairwise generating the finite alternating group

Let $G$ be the alternating group of degree $n$. Let $ω(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $σ(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that, when $n$ varies in the family of composite numbers, $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we explicitly calculate $σ(A_n)$ for $n \geq 21$ congruent to $3$ modulo $18$.

math.GR↗

arXiv2022

On the maximal number of elements pairwise generating the symmetric group of even degree

Let $G$ be the symmetric group of degree $n$. Let $ω(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $σ(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that both functions $σ(G)$ and $ω(G)$ are asymptotically equal to $\frac{1}{2} \binom{n}{n/2}$ when $n$ is even. This, together with a result of S. Blackburn, implies that $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we give a lower bound of $(1-o(1))n$ on $ω(G)$ which is independent of the classification of finite simple groups. We also calculate, for large enough $n$, the clique number of the graph defined as follows: the vertices are the elements of $G$ and two vertices $x,y$ are connected by an edge if $\langle x,y \rangle \geq A_n$.

math.GR↗

arXiv2021

The Fitting height is bounded by a function of the exponent

Every finite solvable group $G$ has a normal series with nilpotent factors. The smallest possible number of factors in such a series is called the Fitting height $h(G)$. In the present paper, we derive an upper bound for $h(G)$ in terms of the exponent of $G$. Our bound constitutes a considerable improvement of an earlier bound obtained by Shalev.

math.GR↗

arXiv2021

On the Primary Coverings of Finite Solvable and Symmetric Groups

A primary covering of a finite group $G$ is a family of proper subgroups of $G$ whose union contains the set of elements of $G$ having order a prime power. We denote with $σ_0(G)$ the smallest size of a primary covering of $G$, and call it the primary covering number of $G$. We study this number and compare it with its analogous $σ(G)$, the covering number, for the classes of groups $G$ that are solvable and symmetric.

math.GR↗

arXiv2019

On the holomorph of finite semisimple groups

Given a finite nonabelian semisimple group $G$, we describe those groups that have the same holomorph as $G$, that is, those regular subgroups $N\simeq G$ of $S(G)$, the group of permutations on the set $G$, such that $N_{S(G)}(N)=N_{S(G)}(ρ(G))$, where $ρ$ is the right regular representation of $G$.

math.GR↗

arXiv2018

A reduction theorem for nonsolvable finite groups

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the nonsolvable length of the group and denoted by $λ(G)$. For every integer $n$, we define a particular class of groups of nonsolvable length $n$, called \emph{$n$-rarefied}, and we show that every finite group of nonsolvable length $n$ contains an $n$-rarefied subgroup. As applications of this result, we improve the known upper bounds on $λ(G)$ and determine the maximum possible nonsolvable length for permutation groups and linear groups of fixed degree resp. dimension.

math.GR↗

arXiv2016

A generalisation of a theorem of Wielandt

In 1974, Helmut Wielandt proved that in a finite group $G$, a subgroup $A$ is subnormal if and only if it is subnormal in every $\seq{A,g}$ for all $g\in G$. In this paper, we prove that the subnormality of an odd order nilpotent subgroup $A$ of $G$ is already guaranteed by a seemingly weaker condition: $A$ is subnormal in $G$ if for every conjugacy class $C$ of $G$ there exists $c\in C$ for which $A$ is subnormal in $\seq{A,c}$. We also prove the following property of finite non-abelian simple groups: if $A$ is a subgroup of odd prime order $p$ in a finite almost simple group $G$, then there exists a cyclic $p'$-subgroup of $F^*(G)$ which does not normalise any non-trivial $p$-subgroup of $G$ that is generated by conjugates of~$A$.

math.GR↗

arXiv2014

The solvability of groups with nilpotent minimal coverings

A covering of a group is a finite set of proper subgroups whose union is the whole group. A covering is minimal if there is no covering of smaller cardinality, and it is nilpotent if all its members are nilpotent subgroups. We complete a proof that every group that has a nilpotent minimal covering is solvable, starting from the previously known result that a minimal counterexample is an almost simple finite group.

math.GR↗

arXiv2014

Truncated Quillen coplexes of p-groups

Let p be an odd prime and let P be a p-group. We examine the order complex of the poset of elementary abelian subgroups of P having order at least p^2. S. Bouc and J. Thévenaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer l, the number of spheres of dimension l in this wedge is controlled by the number of extraspecial subgroups X of P having order p^{2l+3} and satisfying Omega_1(C_P(X))=Z(X). We go on to provide a negative answer to a question raised by Bouc and Thévenaz concerning restrictions on the homology groups of the given complex.

math.GR↗

arXiv2008

Some structural results on the non-abelian tensor square of groups

We study the non-abelian tensor square $G\otimes G$ for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G/G' so that $G\otimes G$ is isomorphic to the direct product of $\nabla(G)$ and the non-abelian exterior square $G\wedge G$. For any group G, we characterize the non-abelian exterior square $G\wedge G$ in terms of a presentation of G. Finally, we apply our results to some classes of groups, such as the classes of free soluble and free nilpotent groups of finite rank, and some classes of finite p-groups.

math.GR↗