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Luca Sabatini

Publications and source records attributed to Luca Sabatini.

At least 19 recordsLinked to original sources
arXiv2026

Cayley graphs of quasirandom groups

A finite group $G$ is $\varepsilon$-quasirandom if all its nontrivial irreducible complex representations have degree at least $|G|^\varepsilon$. Building on recent work of Golsefidy-Srinivas, we prove that expansion in a quasirandom group is controlled by expansion in its simple quotients. As a consequence, we remove the product theorem from the hypotheses of the Bourgain-Gamburd expansion machine. Moreover, we combine this result with crown theory to deduce that $1 + \lfloor \varepsilon^{-1} \rfloor$ random elements give an expander Cayley graph with high probability. Finally, generalizing results of Breuillard-Green-Tao and Pyber-Szab\'o, we prove that the diameter of any connected Cayley graph of a quasirandom group is polylogarithmic.

math.GR↗
arXiv2026

The base size of vertex-transitive cubic graphs

We prove that if $\Gamma$ is a finite connected vertex-transitive cubic graph, then either $|V\Gamma| \le 90$, or $\Gamma$ is a split Praeger--Xu graph, or there exist two vertices $\alpha$ and $\beta$ such that the identity is the only automorphism of $\Gamma$ fixing both $\alpha$ and $\beta$.

math.CO↗
arXiv2026

Diameter bounds for arbitrary finite groups and applications

We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: (1) if $G$ is a finite soluble group of exponent $e$, $\mathrm{diam}(G) \ll e (\log |G|)^8$, (2) anabelian groups with bounded-rank composition factors have polylogarithmic diameter, (3) transitive soluble subgroups of $S_n$ have diameter $\ll n^5$, and (4) Grigorchuk's gap conjecture holds for any finitely generated group acting faithfully on a bounded-degree rooted tree. Additionally, conditional on Babai's conjecture, (5) any transitive permutation group of degree $n$ has diameter bounded by a polynomial in $n$ (a folkloric conjecture), and (6) Grigorchuk's gap conjecture holds for residually finite groups, and thus the conjecture reduces to the simple case.

math.GR↗
arXiv2026

Expanding groups with large diameter

We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set. We answer a question of Pyber and Szab\'o (2013) by exhibiting a sequence of finite groups $G_n$ with $|G_n| \to \infty$ admitting bounded generating sets $X_n,Y_n$ such that $\operatorname{Cay}(G_n,X_n)$ is an expander while $\operatorname{Cay}(G_n,Y_n)$ has super-polylogarithmic diameter. The construction uses the semidirect product $G_n = C_p^{n-1} \rtimes S_n$ with $p$ exponentially large in $n$, and the analysis reduces to bounding some exponential sums of permutational type.

math.GR↗
arXiv2025

Groups that produce expander graphs

We survey the known group properties that a sequence of finite groups or group actions needs to satisfy to admit subsets of bounded cardinality producing expander Cayley or Schreier graphs. We prove that an infinite amenable group and solvable groups of bounded derived length do not produce expander Schreier graphs, generalizing with easier proofs results of Lubotzky and Weiss for Cayley graphs. In particular, the poor expansion properties of a group action cannot in general be detected by looking at the abelian sections or at the representations above the stabilizer of a point.

math.GR↗
arXiv2025

Sylow subgroups for distinct primes and intersection of nilpotent subgroups

Let $G$ be a finite group and let $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. We conjecture that there exists $x \in G$ such that $P_i \cap P_i^x$ is inclusion-minimal in $\{ P_i \cap P_i^g : g \in G\}$ for all $i$. As a first step in this direction, we show that a finite group cannot be covered by (proper) Sylow normalizers for distinct primes. Then we settle the conjecture in two opposite situations: symmetric and alternating groups of large degree and metanilpotent groups of odd order. Applications concerning the intersections of nilpotent subgroups are discussed.

math.GR↗
arXiv2025

Probabilistic construction of some extremal $p$-groups

A $p$-group $G$ is called *ab-maximal* if $|H : H'| < |G:G'|$ for every proper subgroup $H$ of $G$. Similarly, $G$ is called *$d$-maximal* if $d(H) < d(G)$ for every proper subgroup $H$ of $G$, where $d(H)$ is the minimal number of generators of $H$. If $G$ is ab-maximal then $|G:G'| \ge p^3 |G'|$, while if $G$ is $d$-maximal and $p \ne 2$ then $|G:G'| \ge p^2 |G'|$. Answering questions of Gonz\'alez-S\'anchez--Klopsch and Lisi--Sabatini, for all $p$ we construct infinitely many ab-maximal $p$-groups of class $2$ with $|G:G'| = p^3 |G'|$, and infinitely many $d$-maximal $p$-groups of class $2$ with $|G:G'| = p^2 |G'|$. The construction is probabilistic and based on the degeneracy of random alternating bilinear maps on subspaces. It is notable however that in the ab-maximal case we do not have a high-probability result but rather in a suitable sense the proportion of class-$2$ groups with $|G:G'| = p^n$ and $|G'| = p^{n-3}$ that are ab-maximal is close to $1/e$.

math.GR↗
arXiv2024

On stabilizers in finite permutation groups

Let $G$ be a permutation group on the finite set $\Omega$. We prove various results about partitions of $\Omega$ whose stabilizers have good properties. In particular, in every solvable permutation group there is a set-stabilizer whose orbits have length at most $6$, which is best possible and answers two questions of Babai. Every solvable maximal subgroup of any almost simple group has derived length at most $10$, which is best possible. In every primitive group with solvable stabilizer, there are two points whose stabilizer has derived length bounded by an absolute constant.

math.GR↗
arXiv2024

The diameter of random Schreier graphs

We give a combinatorial proof of the following theorem. Let $G$ be any finite group acting transitively on a set of cardinality $n$. If $S \subseteq G$ is a random set of size $k$, with $k \geq (\log n)^{1+\varepsilon}$ for some $\varepsilon >0$, then the diameter of the corresponding Schreier graph is $O(\log_k n)$ with high probability. Except for the implicit constant, this result is the best possible.

math.CO↗
arXiv2024

Quasirandom and quasisimple groups

Fix $\varepsilon > 0$. We say that a finite group $G$ is $\varepsilon$-quasirandom if every nontrivial irreducible complex representation of $G$ has degree at least $|G|^\varepsilon$. In this paper, we give a structure theorem for large $\varepsilon$-quasirandom groups, and we completely classify the $\frac{1}{5}$-quasirandom groups.

math.GR↗
arXiv2024

Fixing two points in primitive solvable groups

Consider a finite primitive solvable group. We observe that a result of Y. Yang implies that there exist two points whose pointwise stabilizer has derived length at most $9$. We show that, if the group has odd cardinality, then there exist two points whose pointwise stabilizer is abelian.

math.GR↗
arXiv2023

On finite $d$-maximal groups

Let $d$ be a positive integer. A finite group is called $d$-maximal if it can be generated by precisely $d$ elements, while its proper subgroups have smaller generating sets. For $d\in\{1,2\}$, the $d$-maximal groups have been classified up to isomorphism and only partial results have been proven for larger $d$. In this work, we prove that a $d$-maximal group is supersolvable and we give a characterization of $d$-maximality in terms of so-called maximal $(p,q)$-pairs. Moreover, we classify the maximal $(p,q)$-pairs of small rank obtaining, as a consequence, a full classification of the isomorphism classes of $3$-maximal finite groups.

math.GR↗
arXiv2022

Products of subgroups, subnormality, and relative orders of elements

Let $G$ be a group. We give an explicit description of the set of elements $x \in G$ such that $x^{|G:H|} \in H$ for every subgroup of finite index $H \leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?

math.GR↗
arXiv2022

On groups with large verbal quotients

Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F = \langle x_1,...,x_n \rangle$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{ w(x_1,...,x_n) : x_1,...,x_n \in G \}$ of all $w$-values in $G$. Following J. Gonz\'alez-S\'anchez and B. Klopsch, a group $G$ is $w$-maximal if $|H:w(H)| < |G:w(G)|$ for every $H<G$. In this paper we give new results on $w$-maximal groups, and study the weaker condition in which the previous inequality is not strict. Some applications are given: for example, if a finite group has a solvable (resp. nilpotent) section of size $n$, then it has a solvable (resp. nilpotent) subgroup of size at least $n$.

math.GR↗
arXiv2022

Nilpotent subgroups of class $2$ in finite groups

We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.

math.GR↗
arXiv2022

Abelian sections of the symmetric groups with respect to their index

We show the existence of an absolute constant $α>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and for every $H \leqslant G$ of index at least $3$, one has $|H/[H,H]| \leq |G:H|^{α/ \log \log |G:H|}$. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.

math.GR↗