Daniele Garzoni

activity

20172026

most citedRandom generation with cycle type restrictions

3 citations · 10 across the 15 of their papers we have counts for

collaborators

16 papers

math.GR2026

On a conjecture of Peter Neumann on fixed points in permutation groups

Daniele Garzoni, Robert M. Guralnick, Martin W. Liebeck

We prove a conjecture of Peter Neumann from 1966, predicting that every finite non-regular primitive permutation group of degree $n$ contains an element fixing at least one point a…

math.NT2025★ 1 cited

Irreducibility of the characteristic polynomials of random tridiagonal matrices

Lior Bary-Soroker, Daniele Garzoni, Sasha Sodin

Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension…

math.GR2024

Derangements in non-Frobenius groups

Daniele Garzoni

We prove that if $G$ is a transitive permutation group of sufficiently large degree $n$ , then either $G$ is primitive and Frobenius, or the proportion of derangements in $G$ is lar…

math.NT2024

On the irreducibility of $f (2^{n}, 3^{m}, X)$ and other such polynomials

Lior Bary-Soroker, Daniele Garzoni, Vlad Matei

Let $f (t_{1}, \dots, t_{r}, X) \in Z [t_{1}, \dots, t_{r}, X]$ be irreducible and let $a_{1}, \dots, a_{r} \in Z ∖ {0, \pm 1}$ . Under a necessary ramification as…

math.GR2024

Probabilistic Generation of Finite Almost Simple Groups

Jason Fulman, Daniele Garzoni, Robert M. Guralnick

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability t…

math.GR2023★ 3 cited

Conjugacy classes of derangements in finite groups of Lie type

Sean Eberhard, Daniele Garzoni

Let $G$ be a finite almost simple group of Lie type acting faithfully and primitively on a set $Ω$ . We prove an analogue of the Boston--Shalev conjecture for conjugacy classes: the…

math.GR2022

Probability of generation by random permutations of given cycle type

Sean Eberhard, Daniele Garzoni

Suppose $π$ and $π^{'}$ are two random elements of $S_{n}$ with constrained cycle types such that $π$ has $x n^{1/2}$ fixed points and $y n /2$ two-cycles, and likewise $π^{'}$ has $x' n^{1/…

math.GR2022

Minimal invariable generating sets

Daniele Garzoni, Andrea Lucchini

A subset $S$ of a group $G$ invariably generates $G$ if, when each element of $S$ is replaced by an arbitrary conjugate, the resulting set generates $G .$ An invariable generating s…

math.NT2022

Hilbert's irreducibility theorem via random walks

Lior Bary-Soroker, Daniele Garzoni

Let $G$ be a connected linear algebraic group over a number field $K$ , let $Γ$ be a finitely generated Zariski dense subgroup of $G (K)$ and let $Z \subseteq G (K)$ be a thin set, in…

math.GR2020

Large minimal invariable generating sets in the finite symmetric groups

Daniele Garzoni, Nick Gill

For a finite group $G$ , let $m_{I} (G)$ denote the largest possible cardinality of a minimal invariable generating set of $G$ . We prove an upper and a lower bound for $m_{I} (S_{n})$ , whic…

math.GR2020★ 1 cited

On the number of conjugacy classes of a primitive permutation group with nonabelian socle

Daniele Garzoni, Nick Gill

Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k (G)$ be the number of conjugacy classes of $G$ . We prove that either $k (G) < n /2$ and $k(G)=o(…

math.GR2020

Connected components in the invariably generating graph of a finite group

Daniele Garzoni

We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.

math.GR2020★ 1 cited

On the probability of generating invariably a finite simple group

Daniele Garzoni, Eilidh McKemmie

Let $G$ be a finite simple group. In this paper we consider the existence of small subsets $A$ of $G$ with the property that, if $y \in G$ is chosen uniformly at random, then with…

math.CO2019★ 3 cited

Random generation with cycle type restrictions

Sean Eberhard, Daniele Garzoni

We study random generation in the symmetric group when cycle type restrictions are imposed. Given $π, π^{'} \in S_{n}$ , we prove that $π$ and a random conjugate of $π^{'}$ are likely to ge…

math.GR2018★ 1 cited

Generating alternating and symmetric groups with two elements of fixed order

Daniele Garzoni

We answer a question raised by Lanier about the possibility of generating $A_{n}$ and $S_{n}$ with two elements of order $k$ , where $n ⩾ k ⩾ 3$ . We show that this can a…

math.GR2017

The invariably generating graph of the alternating and symmetric groups

Daniele Garzoni

Given a finite group $G$ , the invariably generating graph of $G$ is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of $G$ , and two class…

On a conjecture of Peter Neumann on fixed points in permutation groups

Irreducibility of the characteristic polynomials of random tridiagonal matrices

Derangements in non-Frobenius groups

On the irreducibility of f(2n,3m,X) and other such polynomials

Probabilistic Generation of Finite Almost Simple Groups

Conjugacy classes of derangements in finite groups of Lie type

Probability of generation by random permutations of given cycle type

Minimal invariable generating sets

Hilbert's irreducibility theorem via random walks

Large minimal invariable generating sets in the finite symmetric groups

On the number of conjugacy classes of a primitive permutation group with nonabelian socle

Connected components in the invariably generating graph of a finite group

On the probability of generating invariably a finite simple group

Random generation with cycle type restrictions

Generating alternating and symmetric groups with two elements of fixed order

The invariably generating graph of the alternating and symmetric groups

On the irreducibility of $f (2^{n}, 3^{m}, X)$ and other such polynomials