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.

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| →∞ admitting bounded generating sets X_n,Y_n such that Cay(G_n,X_n) is an expander while Cay(G_n,Y_n) has super-polylogarithmic diameter. The construction uses the semidirect product G_n = C_p^n-1⋊ S_n with p exponentially large in n, and the analysis reduces to bounding some exponential sums of permutational type.

We compute the asymptotic probability that a random pair of Sylow 2-subgroups in S n S_{n} or A n A_{n} intersects trivially. This calculation complements recent work of Diaconis, Giannelli, Guralnick, Law, Navarro, Sambale, and Spink.

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).

A p-group G is called ab-marimal if |H: H'| = p <^> 9|G'| while if G is d-maximal and p 2 then |G / (G')| = p(2)|G'| Answering questions of Gonz & aacute;lez-S & aacute;nchez-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 |G / (G')| = p(2)|G'| The construction is probabilistic and based 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 (where e is the base of the natural logarithm). (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

We give a description of non-growing subsets in linear groups, which extends the Product theorem for simple groups of Lie type. We also give an account of various related aspects of growth in linear groups.

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 $A_n$.

Let G be a finite classical group generated by transvections, i.e., one of SLn(q), SUn(q), Sp2n(q), or O2n±(q)(qeven), 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 Cay(G,X) is bounded by (nlog⁡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 SLn(q), SUn(q), Sp2n(q) and X contains three random generators then with high probability the diameter Cay(G,X) is bounded by nO(log⁡q). This confirms Babai's conjecture for non-orthogonal classical simple groups over small fields and three random generators.

The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essentially no new examples. There are many further examples arising from nondesarguesian planes. We conjecture that all dense Sidon sets arise from finite projective planes in this way. If true, this implies that all abelian groups of most orders do not have dense Sidon subsets. In particular if $\sigma_n$ denotes the size of the largest Sidon subset of $\mathbb{Z}/n\mathbb{Z}$, this implies $\liminf_{n \to \infty} \sigma_n / n^{1/2} < 1$. We also give a brief bestiary of somewhat smaller Sidon sets with a variety of algebraic origins, and for some of them provide an overarching pattern.

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 $d$.

For G a finite group, let d_2(G) denote the proportion of triples (x, y, z) ∈ G^3 such that [x, y, z] = 1 . We determine the structure of finite groups G such that d_2(G) is bounded away from zero: if d_2(G) ≥ϵ > 0 , G has a class-4 nilpotent normal subgroup H such that [G : H] and |γ _4(H)| are both bounded in terms of ϵ . We also show that if G is an infinite group whose commutators have boundedly many conjugates, or indeed if G satisfies a certain more general commutator covering condition, then G is finite-by-class-3-nilpotent-by-finite.

A transversal in an nxn$n \times n$ latin square is a collection of n$n$ entries not repeating any row, column, or symbol. Kwan showed that almost every nxn$n \times n$ latin square has (1+o(1))n/e2n$\bigl ((1 + o(1)) n / e<^>2\bigr )<^>n$ transversals as n ->infinity$n \rightarrow \infty$. Using a loose variant of the circle method we sharpen this to (e-1/2+o(1))n!2/nn$(e<^>{-1/2} + o(1)) n!<^>2 / n<^>n$. 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.

We describe primitive association schemes 𝔛 of degree n such that Aut(𝔛) is imprimitive and |Aut(𝔛)| ≥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 ≤ 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^1/8log n) automorphisms.

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 = ⟨π, π' ⟩ is transitive and G ≥ A_n both have probability approximately (1 - yy')^1/2exp(- xx' + 1/2 x^2 y' + 1/2x'^2 y/1 - yy'), provided (x, x') is not close to (0, ∞) or (∞, 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

Hall and Paige conjectured in 1955 that a finite group G has a complete mapping if and only if its Sylow 2-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group G of order n satisfying the Hall–Paige condition is (e−1/2+o(1))|Gab|n!2/nn.

Form an n × n matrix by drawing entries independently from {±1} (or another fixed nontrivial finitely supported distribution in Z ) and let φ be the characteristic polynomial. We show, conditionally on the extended Riemann hypothesis, that with high probability φ is irreducible and Gal( φ ) ≥ A n .

We adapt the theory of partition rank and analytic rank to the category of abelian groups. If A_1, …, A_k are finite abelian groups and ϕ : A_1 ×⋯× A_k →𝐓 is a multilinear map, where 𝐓 = 𝐑/𝐙, the bias of ϕ is defined to be the average value of exp(i 2 πϕ). If the bias of ϕ is bounded away from zero we show that ϕ is the sum of boundedly many multilinear maps each of which factors through the standard multiplication map of 𝐙/q𝐙 for some bounded prime power q. Relatedly, if F : A_1 ×⋯× A_k-1→ B is a multilinear map such that 𝐏(F = 0) is bounded away from zero, we show that F is the sum of boundedly many multilinear functions of a particular form. These structure theorems generalize work of several authors in the elementary abelian case to the arbitrary abelian case. The set of all possible biases is also investigated.

A family of vectors in [ k ] n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [ k ] n invariant under a transitive group of symmetries is o ( k n ), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.

Let $G = \mathrm{SCl}_n(q)$ be a quasisimple classical group with $n$ large, and let $x_1, \dots, x_k \in G$ random, where $k \geq q^C$. We show that the diameter of the resulting Cayley graph is bounded by $q^2 n^{O(1)}$ with probability $1 - o(1)$. In the particular case $G = \mathrm{SL}_n(p)$ with $p$ a prime of bounded size, we show that the same holds for $k = 3$.

We study random generation in the symmetric group when cycle type restrictions are imposed. Given π, π' ∈ S_n, we prove that π and a random conjugate of π' are likely to generate at least A_n provided only that π and π' have not too many fixed points and not too many 2-cycles. As an application, we investigate the following question: For which positive integers m should we expect two random elements of order m to generate A_n? Among other things, we give a positive answer for any m having any divisor d in the range 3 ≤ d ≤ o(n^1/2).