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Fabio Mastrogiacomo

Publications and source records attributed to Fabio Mastrogiacomo.

7 recordsLinked to original sources
arXiv2026

Finite groups in which every proper characteristic subgroup is cyclic

Let $G$ be a finite non-cyclic, non-characteristically simple group with the property that all proper characteristic subgroups of $G$ are cyclic. We call such a group $\mathrm{CCS}$ group, short for \emph{Characteristic Cyclic}. In this paper, we provide a complete classification of these groups. As an application of our main result, we also make some progress toward the classification of minimal non-cyclic skew braces.

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arXiv2025

On the base size and minimal degree of transitive groups

Let $G$ be a permutation group, and denote with $μ(G)$ and $b(G)$ its minimal degree and base size respectively. We show that for every $\varepsilon>0$, there exists a transitive permutation group $G$ of degree $n$ with \[ μ(G)b(G) \geq n^{2-\varepsilon}. \] We also identify some classes of transitive and intransitive groups whose base size and minimal degree have a smaller upper bound, shared with primitive groups.

math.GR↗
arXiv2025

IBIS primitive groups of almost simple type

Let $G$ be a finite permutation group on $Ω$. An ordered sequence $(ω_1\ldots,ω_\ell)$ of elements of $Ω$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. The minimal cardinality of a base is said to be the base size of $G$. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. In this paper, we classify the finite almost simple primitive IBIS groups whose base size is at least $6$.

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arXiv2024

On the minimal degree and base size of finite primitive groups

Let $G$ be a finite permutation group acting on $Ω$. A base for $G$ is a subset $B \subseteq Ω$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the smallest possible base. The minimal degree of $G$, denoted by $μ(G)$, is the smallest cardinality of the support of a non trivial element of $G$. In this paper, we establish a new upper bound for $b(G)$ when $G$ is primitive, and subsequently prove that if $G$ is a primitive group different from the Mathieu group of degree $24$, then $μ(G)b(G)\leq n \log n$, where $n$ is the degree of $G$. This bound is best possible, up to a multiplicative constant.

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arXiv2024

Cardinalities of irredundant bases of finite primitive groups

Let $G$ be a finite permutation group acting on a set $Ω$. An ordered sequence $(ω_1,\ldots,ω_\ell)$ of elements of $Ω$ is an irredundant base for $G$ if the pointwise stabilizer of the sequence is trivial and no point is fixed by the stabilizer of its predecessors. We show that any interval of natural numbers can be realized as the set of cardinalities of irredundant bases for some finite primitive group.

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arXiv2023

On the cardinality of irredundant and minimal bases of finite permutation groups

Given a finite permutation group $G$ with domain $Ω$, we associate two subsets of natural numbers to $G$, namely $\mathcal{I}(G,Ω)$ and $\mathcal{M}(G,Ω)$, which are the sets of cardinalities of all the irredundant and minimal bases of $G$, respectively. We prove that $\mathcal{I}(G)$ is an interval of natural numbers, whereas $\mathcal{M}(G,Ω)$ may not necessarily form an interval. Moreover, for a given subset of natural numbers $X \subseteq \mathbb{N}$, we provide some conditions on $X$ that ensure the existence of both intransitive and transitive groups $G$ such that $\mathcal{I}(G,Ω) = X$ and $\mathcal{M}(G,Ω) = X$.

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