Sean Eberhard
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Sean Eberhard has a total of 20 co-authors. Here are the top co-authors by number of shared publications:
- Freddie Manners
UCSD
4 shared publications - Daniele Garzoni
Department of Mathematics University of Southern California 90089-2532 Los Angeles CA USA
3 shared publications - Péter Pál Varjú
Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK
2 shared publications - Rudi Mrazović
Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia
2 shared publications - Brendan Murphy
School of Mathematics, University of Bristol, Bristol, UK
1 shared publication - Charles Burnette
Xavier University of Louisiana
1 shared publication
Latest Publications
- Open Access
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 Szabó (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.Open AccessExpanding 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 Szabó (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. - Open Access
- 31 October 2025
- Published by Elsevier in Journal of Number Theory
- Vol. 281, 426-428
Ratios of consecutive values of the divisor function
- 31 October 2025
- Published by Elsevier in Journal of Number Theory
- Vol. 281, 426-428
Open AccessRatios of consecutive values of the divisor function
- Open Access
- 14 October 2025
- Published by Walter de Gruyter GmbH in Journal of Group Theory
Intersections of Sylow 2-subgroups in symmetric groups
We compute the asymptotic probability that a random pair of Sylow 2-subgroups in or intersects trivially. This calculation complements recent work of Diaconis, Giannelli, Guralnick, Law, Navarro, Sambale, and Spink.- 14 October 2025
- Published by Walter de Gruyter GmbH in Journal of Group Theory
Open AccessIntersections of Sylow 2-subgroups in symmetric groups
We compute the asymptotic probability that a random pair of Sylow 2-subgroups in or intersects trivially. This calculation complements recent work of Diaconis, Giannelli, Guralnick, Law, Navarro, Sambale, and Spink. - Open AccessExamples of diameter-2 graphs with no triangle or $K_{2,t}$For each $t \ge 1$ let $W_t$ denote the class of graphs other than stars that have diameter $2$ and contain neither a triangle nor a $K_{2,t}$. The famous Hoffman--Singleton Theorem implies that $W_2$ is finite. Recently Wood suggested the study of $W_t$ for $t > 2$ and conjectured that $W_t$ is finite for all $t \ge 2$. In this note we show that (1) $W_3$ is infinite, (2) $W_5$ contains infinitely many regular graphs, and (3) $W_7$ contains infinitely many Cayley graphs. Our $W_3$ and $W_5$ examples are based on so-called crooked graphs, first constructed by de Caen, Mathon, and Moorhouse. Our $W_7$ examples are Cayley graphs with vertex set $\mathbb{F}_p^2$ for prime $p \equiv 11 \pmod {12}$.Open AccessExamples of diameter-2 graphs with no triangle or $K_{2,t}$For each $t \ge 1$ let $W_t$ denote the class of graphs other than stars that have diameter $2$ and contain neither a triangle nor a $K_{2,t}$. The famous Hoffman--Singleton Theorem implies that $W_2$ is finite. Recently Wood suggested the study of $W_t$ for $t > 2$ and conjectured that $W_t$ is finite for all $t \ge 2$. In this note we show that (1) $W_3$ is infinite, (2) $W_5$ contains infinitely many regular graphs, and (3) $W_7$ contains infinitely many Cayley graphs. Our $W_3$ and $W_5$ examples are based on so-called crooked graphs, first constructed by de Caen, Mathon, and Moorhouse. Our $W_7$ examples are Cayley graphs with vertex set $\mathbb{F}_p^2$ for prime $p \equiv 11 \pmod {12}$.
- Open Access
- 28 July 2025
- Published by Wiley in Bulletin of the London Mathematical Society
- Vol. 57 (11) , 3307-3325
Normal covering numbers for Sn$S_n$ and An$A_n$ and additive combinatorics
The normal covering number of a noncyclic group is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for and depending on the arithmetic structure of . In particular we determine the limsups over and over the sequences of even and odd integers, as well as the liminf of over even integers. In general we explain how the values of and are related to problems in additive combinatorics. These results answer most of the questions posed by Bubboloni, Praeger, and Spiga as Problem 20.17 of the Kourovka Notebook.
- 28 July 2025
- Published by Wiley in Bulletin of the London Mathematical Society
- Vol. 57 (11) , 3307-3325
Open AccessNormal covering numbers for Sn$S_n$ and An$A_n$ and additive combinatorics
The normal covering number of a noncyclic group is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for and depending on the arithmetic structure of . In particular we determine the limsups over and over the sequences of even and odd integers, as well as the liminf of over even integers. In general we explain how the values of and are related to problems in additive combinatorics. These results answer most of the questions posed by Bubboloni, Praeger, and Spiga as Problem 20.17 of the Kourovka Notebook.
- Open Access
- 23 June 2025
- Published by Elsevier in Journal of Algebra
- Vol. 682, 463-480
Probabilistic construction of some extremal p-groups
- 23 June 2025
- Published by Elsevier in Journal of Algebra
- Vol. 682, 463-480
Open AccessProbabilistic construction of some extremal p-groups
- Open Access
- 22 April 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 12 (16) , 536-575
Conjugacy classes of derangements in finite groups of Lie type
Let G G be a finite almost simple group of Lie type acting faithfully and primitively on a set Ω \Omega . We prove an analogue of the Boston–Shalev conjecture for conjugacy classes: the proportion of conjugacy classes of G G consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry).
- 22 April 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 12 (16) , 536-575
Open AccessConjugacy classes of derangements in finite groups of Lie type
Let G G be a finite almost simple group of Lie type acting faithfully and primitively on a set Ω \Omega . We prove an analogue of the Boston–Shalev conjecture for conjugacy classes: the proportion of conjugacy classes of G G consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry).
- Open Access
- 1 January 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 12 (16) , 536-575
Conjugacy classes of derangements in finite groups of Lie type
Let $G$ be a finite almost simple group of Lie type acting faithfully and primitively on a set $\Omega$. We prove an analogue of the Boston–Shalev conjecture for conjugacy classes: the proportion of conjugacy classes of $G$ consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry).- 1 January 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 12 (16) , 536-575
Open AccessConjugacy classes of derangements in finite groups of Lie type
Let $G$ be a finite almost simple group of Lie type acting faithfully and primitively on a set $\Omega$. We prove an analogue of the Boston–Shalev conjecture for conjugacy classes: the proportion of conjugacy classes of $G$ consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry). - Open Access
- 10 May 2024
- Published by Elsevier in Journal of Algebra
- Vol. 653, 220-256
Diameter of classical groups generated by transvections
- 10 May 2024
- Published by Elsevier in Journal of Algebra
- Vol. 653, 220-256
Open AccessDiameter of classical groups generated by transvections
- Open Access
Growth in linear groups
We prove a conjecture of Helfgott on the structure of sets of bounded tripling in bounded rank, which states the following. Let $A$ be a finite symmetric subset of $\mathrm{GL}_n(\mathbf{F})$ for any field $\mathbf{F}$ such that $|A^3| \leq K|A|$. Then there are subgroups $H \trianglelefteq \Gamma \trianglelefteq \langle A \rangle$ such that $A$ is covered by $K^{O_n(1)}$ cosets of $\Gamma$, $\Gamma/H$ is nilpotent of step at most $n-1$, and $H$ is contained in $A^{O_n(1)}$. This theorem includes the Product Theorem for finite simple groups of bounded rank as a special case. As an application of our methods we also show that the diameter of sufficiently quasirandom finite linear groups is poly-logarithmic.Open AccessGrowth in linear groups
We prove a conjecture of Helfgott on the structure of sets of bounded tripling in bounded rank, which states the following. Let $A$ be a finite symmetric subset of $\mathrm{GL}_n(\mathbf{F})$ for any field $\mathbf{F}$ such that $|A^3| \leq K|A|$. Then there are subgroups $H \trianglelefteq \Gamma \trianglelefteq \langle A \rangle$ such that $A$ is covered by $K^{O_n(1)}$ cosets of $\Gamma$, $\Gamma/H$ is nilpotent of step at most $n-1$, and $H$ is contained in $A^{O_n(1)}$. This theorem includes the Product Theorem for finite simple groups of bounded rank as a special case. As an application of our methods we also show that the diameter of sufficiently quasirandom finite linear groups is poly-logarithmic.