Luca Sabatini

activity

20202022

2

in finite groups

4 citations · 12 across the 8 of their papers we have counts for

collaborators

8 papers

math.GR2022★ 1 cited

Products of subgroups, subnormality, and relative orders of elements

Luca Sabatini

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 $H ⩽ G$ . This is related…

math.GR2022★ 1 cited

On groups with large verbal quotients

Francesca Lisi, Luca Sabatini

Let $w = w (x_{1}, ..., x_{n})$ be a word, i.e. an element of the free group $F = ⟨ x_{1}, ..., x_{n} ⟩$ . The verbal subgroup $w (G)$ of a group $G$ is the subgroup generated by the se…

math.GR2021★ 4 cited

Nilpotent subgroups of class $2$ in finite groups

Luca Sabatini

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 $∣ G ∣^{1/32 l o g l o g ∣ G ∣}$ . This answers a question of Pyber,…

math.GR2021★ 1 cited

Abelian sections of the symmetric groups with respect to their index

Luca Sabatini

We show the existence of an absolute constant $α > 0$ such that, for every $k \geq 3$ , $G := Sym (k)$ , and for every $H ⩽ G$ of index at least $3$ , one has $|H…

math.GR2021

On finite groups with polynomial diameter

Luca Sabatini

Given a finite group $G$ and a generating set $S \subseteq G$ , the diameter $d iam (G, S)$ is the least integer $n$ such that every element of $G$ is the product of at most $n$ elemen…

math.GR2021★ 2 cited

The growth of abelian sections

Luca Sabatini

Given an abstract group $G$ , we study the function $a b_{n} (G) := sup_{∣ G : H ∣ \leq n} ∣ H / [H, H] ∣$ . If $G$ has no abelian composition factors, then $a b_{n} (G)$ is bounded by a polynomial:…

math.CO2021★ 3 cited

Random Schreier graphs and expanders

Luca Sabatini

Let the group $G$ act transitively on the finite set $Ω$ , and let $S \subseteq G$ be closed under taking inverses. The Schreier graph $S c h (G ↺ Ω, S)$ is the graph wit…

math.GR2020

A subexponential bound on the cardinality of abelian quotients in finite transitive groups

Andrea Lucchini, Luca Sabatini, Pablo Spiga

We show that, for every transitive group $G$ of degree $n \geq 2$ , the largest abelian quotient of $G$ has cardinality at most $4^{n / l o g_{2} n}$ . This gives a positive answer t…

Products of subgroups, subnormality, and relative orders of elements

On groups with large verbal quotients

Nilpotent subgroups of class 2 in finite groups

Abelian sections of the symmetric groups with respect to their index

On finite groups with polynomial diameter

The growth of abelian sections

Random Schreier graphs and expanders

A subexponential bound on the cardinality of abelian quotients in finite transitive groups

Nilpotent subgroups of class $2$ in finite groups