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Sean Eberhard

Publications and source records attributed to Sean Eberhard.

At least 19 recordsLinked to original sources
arXiv2026

Forcible groups and Frattini covers

We say that a finite group $G$ *forces* a finite group $H$ if every finite cover of $G$ contains a subgroup isomorphic to $H$, and we say $H$ is *forcible* if some finite group $G$ forces $H$. Complementing a classical result of Thompson--Mann, and answering a recent question of the third author, we show that a finite group is forcible if and only if it is abelian and its Sylow subgroups are elementary-by-cyclic. We also prove relative forcibility results for abelian $p$-groups in the settings of powerful $p$-groups and $p$-groups of bounded nilpotency class.

math.GR↗
arXiv2026

Fixed-point-free elements in two-orbit permutation groups

Let $G$ be a two-orbit permutation group on $n > 2$ points. We show that $G$ contains either a derangement or an element of prime-power order with a unique fixed point. As a corollary, if the orbits of $G$ have length $n_1$ and $n_2$ and $\gcd(n_1, n_2-1) = \gcd(n_1-1, n_2) = 1$, then $G$ contains a derangement. The special case $n_1 = n_2$ was recently conjectured by Ellis and Harper and proved under various restrictive hypotheses. We prove our result by reducing to the case of simple groups and leveraging the classification of normal $2$-coverings of simple groups due to Bubboloni, Spiga, and Weigel.

math.GR↗
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

The growth of residually soluble groups

Building on work of Wilson, we show that if $G$ is a finitely generated residually soluble group whose growth function $\gamma$ satisfies $(\log \gamma(n))/ n^{1/4} \to 0$ as $n \to \infty$ then $G$ is virtually nilpotent. This shows that Grigorchuk's Gap Conjecture holds for all exponents $\beta < 1/4$ within the class of residually soluble groups (improving Wilson's exponent $1/6$). We also discuss stronger versions of the Gap Conjecture.

math.GR↗
arXiv2025

Examples 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 $p \equiv 11 \pmod {12}$.

math.CO↗
arXiv2025

Intersections of Sylow 2-subgroups in symmetric groups

We compute the asymptotic probability that a random pair of Sylow 2-subgroups in $S_n$ or $A_n$ intersects trivially. This calculation complements recent work of Diaconis, Giannelli, Guralnick, Law, Navarro, Sambale, and Spink (see arXiv:2504.01149).

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

Normal covering numbers for $S_n$ and $A_n$ and additive combinatorics

The normal covering number $\gamma(G)$ of a finite group $G$ is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for $\gamma(S_n)$ and $\gamma(A_n)$ depending on the arithmetic structure of $n$. In particular we determine the limsups over $\gamma(S_n) / n$ and $\gamma(A_n) / n$ over the sequences of even and odd integers, as well as the liminf of $\gamma(S_n) / n$ over even integers. In general we explain how the values of $\gamma(S_n) / n$ and $\gamma(A_n) / n$ 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.

math.GR↗
arXiv2023

Diameter of classical groups generated by transvections

Let $G$ be a finite classical group generated by transvections, i.e., one of $\operatorname{SL}_n(q)$, $\operatorname{SU}_n(q)$, $\operatorname{Sp}_{2n}(q)$, or $\operatorname{O}^\pm_{2n}(q)$ ($q$ even), and let $X$ be a generating set for $G$ containing at least one transvection. Building on work of Garonzi, Halasi, and Somlai, we prove that the diameter of the Cayley graph $\operatorname{Cay}(G, X)$ is bounded by $(n \log q)^C$ for some constant $C$. This confirms Babai's conjecture on the diameter of finite simple groups in the case of generating sets containing a transvection. By combining this with a result of the author and Jezernik it follows that if $G$ is one of $\operatorname{SL}_n(q)$, $\operatorname{SU}_n(q)$, $\operatorname{Sp}_{2n}(q)$ and $X$ contains three random generators then with high probability the diameter $\operatorname{Cay}(G, X)$ is bounded by $n^{O(\log q)}$. This confirms Babai's conjecture for non-orthogonal classical simple groups over small fields and three random generators.

math.GR↗
arXiv2023

Dixon's asymptotic without CFSG

Without using the classification of finite simple groups, we show that the probability that two random elements of $S_n$ generate a primitive group smaller than $A_n$ is at most $\exp(-c(n \log n)^{1/2})$. As a corollary we get Dixon's asymptotic expansion \[ 1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - \cdots \] for the probability that two random elements of $S_n$ (or $A_n$) generate a subgroup containing $A_n$.

math.GR↗
arXiv2023

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).

math.GR↗
arXiv2022

Transversals in quasirandom latin squares

A transversal in an $n \times n$ latin square is a collection of $n$ entries not repeating any row, column, or symbol. Kwan showed that almost every $n \times n$ latin square has $\bigl((1 + o(1)) n / e^2\bigr)^n$ transversals as $n \to \infty$. Using a loose variant of the circle method we sharpen this to $(e^{-1/2} + o(1)) n!^2 / n^n$. Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.

math.CO↗
arXiv2022

Fusions of tensor powers of Johnson schemes

This paper is a follow-up to (arXiv:2203.03687), in which the first author studied primitive association schemes lying between a tensor power $\mathcal{T}_m^d$ of the trivial association scheme and the Hamming scheme $\mathcal{H}(m,d)$. A question which arose naturally in that study was whether all primitive fusions of $\mathcal{T}_m^d$ lie between $\mathcal{T}_{m^e}^{d/e}$ and $\mathcal{H}(m^d, d/e)$ for some $e \mid d$. This note answers this question positively provided that $m$ is large enough. We similarly classify primitive fusions of the $d$th tensor power of a Johnson scheme on $\binom{m}{k}$ points provided $m$ is large enough in terms of $k$ and $d$.

math.CO↗
arXiv2022

Probability of generation by random permutations of given cycle type

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 $yn/2$ two-cycles, and likewise $π'$ has $x' n^{1/2}$ fixed points and $y'n/2$ two-cycles. We show that the events that $G = \langle π, π' \rangle$ is transitive and $G \geq A_n$ both have probability approximately \[(1 - yy')^{1/2} \exp\left(- \frac{xx' + \frac12 x^2 y' + \frac12 {x'}^2 y}{1 - yy'}\right),\] provided $(x, x')$ is not close to $(0, \infty)$ or $(\infty, 0)$. This formula is derived from some preliminary results in a recent paper (arXiv:1904.12180) of the authors. As an application, we show that two uniformly random elements of uniformly random conjugacy classes of $S_n$ generate the group with probability about 51%.

math.GR↗
arXiv2022

Hamming sandwiches

We describe primitive association schemes $\mathfrak{X}$ of degree $n$ such that $\mathrm{Aut}(\mathfrak{X})$ is imprimitive and $|\mathrm{Aut}(\mathfrak{X})| \geq \exp(n^{1/8})$, contradicting a conjecture of Babai. This and other examples we give are the first known examples of nonschurian primitive coherent configurations (PCC) with more than a quasipolynomial number of automorphisms. Our constructions are "Hamming sandwiches", association schemes sandwiched between the $d$th tensor power of the trivial scheme and the $d$-dimensional Hamming scheme. We study Hamming sandwiches in general, and exhaustively for $d \leq 8$. We revise Babai's conjecture by suggesting that any PCC with more than a quasipolynomial number of automorphisms must be an association scheme sandwiched between a tensor power of a Johnson scheme and the corresponding full Cameron scheme. If true, it follows that any nonschurian PCC has at most $\exp O(n^{1/8} \log n)$ automorphisms.

math.CO↗
arXiv2021

The characteristic polynomial of a random matrix

Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $ϕ$ be the characteristic polynomial. Conditionally on the extended Riemann hypothesis, with high probability $ϕ$ is irreducible and $\mathrm{Gal}(ϕ) \geq A_n$.

math.NT↗