## Publications

150 results found

Liebeck M, Simion I, 2023, Covering numbers for simple algebraic groups, *Vietnam Journal of Mathematics*, Vol: 51, Pages: 605-616, ISSN: 0866-7179

Let G be a simple algebraic group over an algebraically closed field, and let C be a noncentral conjugacy class of G. The covering number cn(G, C) is defined to be the minimal k such that G = Ck , where Ck = {c1c2 · · · ck : ci ∈ C}. We prove that cn(G, C) ≤ c dim G dim C , where c is an explicit constant (at most 120). Some consequences on the width and generation of simple algebraic groups are given.

Liebeck M, 2022, A bound for the orders of centralizers of irreducible subgroups of algebraic groups, *Journal of Group Theory*, Vol: 26, Pages: 795-801, ISSN: 1433-5883

We prove that if 𝐺 is a connected semisimple algebraic group of rank 𝑟, and 𝐻 is a subgroup of 𝐺 that is contained in no proper parabolic subgroup, then we have |CG(H)|<cr|Z(G)| , where 𝑐 is an absolute constant ( c=16 if all simple factors of 𝐺 are classical, and c≤197 in general).

Liebeck M, Seitz G, Testerman D, 2022, Multiplicity-free representations of algebraic groups II, *Journal of Algebra*, Vol: 607, Pages: 531-606, ISSN: 0021-8693

We continue our work, started in [9], on the program of classifying triples(X, Y, V ), where X, Y are simple algebraic groups over an algebraically closed field ofcharacteristic zero with X < Y , and V is an irreducible module for Y such that therestriction V ↓ X is multiplicity-free. In this paper we handle the case where X is oftype A, and is irreducibly embedded in Y of type B, C or D. It turns out that thereare relatively few triples for X of arbitrary rank, but a number of interesting exceptionalexamples arise for small ranks.

Liebeck MW, Praeger CE, 2022, OBITUARY Peter Michael Neumann, 1940-2020, *BULLETIN OF THE LONDON MATHEMATICAL SOCIETY*, Vol: 54, Pages: 1487-1514, ISSN: 0024-6093

Liebeck M, Gill N, Spiga P, 2022, Cherlin's conjecture on finite primitive binary permutation groups, Publisher: Springer Verlag, ISBN: 978-3-030-95955-5

This book gives a proof of Cherlin’s conjecture for finite binary primitive permutation groups. Motivated by the part of model theory concerned with Lachlan’s theory of finite homogeneous relational structures, this conjecture proposes a classification of those finite primitive permutation groups that have relational complexity equal to 2. The first part gives a full introduction to Cherlin’s conjecture, including all the key ideas that have been used in the literature to prove some of its special cases. The second part completes the proof by dealing with primitive permutation groups that are almost simple with socle a group of Lie type. A great deal of material concerning properties of primitive permutation groups and almost simple groups is included, and new ideas are introduced. Addressing a hot topic which cuts across the disciplines of group theory, model theory and logic, this book will be of interest to a wide range of readers. It will be particularly useful for graduate students and researchers who need to work with simple groups of Lie type.

Liebeck M, Tiep PH, 2021, Character ratios for exceptional groups of Lie type, *International Mathematics Research Notices*, Vol: 2021, Pages: 12054-12076, ISSN: 1073-7928

We prove character ratio bounds for finite exceptional groups G(q) of Lie type. These take the form |χ(g)|χ(1)≤cqk for all nontrivial irreducible characters χ and nonidentity elements g, where c is an absolute constant, and k is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs.

Liebeck M, Shalev A, Tiep PH, 2021, McKay graphs for alternating and classical groups, *Transactions of the American Mathematical Society*, Vol: 374, Pages: 5651-5676, ISSN: 0002-9947

Let G be a finite group, andαa nontrivial character of G. The McKay graph M (G,α) has the irreducible characters of Gas vertices, with an edge fromχ1toχ2ifχ2is a constituent ofαχ1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G=An, we prove a conjecture made in [20]: there is an absolute constant C such that diam M (G,α)≤ C log | G| log α (1)for all nontrivial irreducible characters α of G. Also for classical groups of symplectic or orthogonal type of rank r, we establish a linear upper bound Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a productχ1χ2···χlof irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents.

Liebeck M, Shalev A, Tiep PH, 2021, On the diameters of McKay graphs for finite simple groups, *Israel Journal of Mathematics*, Vol: 241, Pages: 449-464, ISSN: 0021-2172

Let Gbe a finite group, andαa nontrivial character of G. The McKay graph M(G,α) has the irreducible characters of Gas vertices, with an edge from χ1 to χ2 if χ2 is a constituent of αχ1. We study the diameters of McKay graphs for simple groups G of Lie type. We show that for anyα, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for G= PSLn(q) or PSUn(q)

Liebeck MW, Shalev A, Tiep PH, 2020, Character ratios, representation varieties and random generation of finite groups of Lie type, *Advances in Mathematics*, Vol: 374, Pages: 1-39, ISSN: 0001-8708

We use character theory of finite groups of Lie type to establish new results on representation varieties of Fuchsian groups, and also on probabilistic generation of groups of Lie type.

Benkart G, Diaconis P, Liebeck M,
et al., 2020, Tensor product Markov chains, *Journal of Algebra*, Vol: 561, Pages: 17-83, ISSN: 0021-8693

We analyze families of Markov chains that arise from decomposing ten-sor products of irreducible representations. This illuminates the Burnside-Brauer Theorem for building irreducible representations, the McKay Corre-spondence, and Pitman’s2M−XTheorem. The chains are explicitly di-agonalizable, and we use the eigenvalues/eigenvectors to give sharp rates ofconvergence for the associated random walks. For modular representations,the chains are not reversible, and the analytical details are surprisingly intri-cate. In the quantum group case, the chains fail to be diagonalizable, but anovel analysis using generalized eigenvectors proves successful.

Liebeck M, Green H, 2020, Some codes in symmetric and linear groups, *Discrete Mathematics*, Vol: 343, ISSN: 0012-365X

For a finite group G, a positive integer λ, and subsets X, Y of G, write λG = XY if the products xy (x ∈ X, y ∈ Y ), cover G precisely λ times. Such a subset Y is called a code with respect to X, and when λ = 1 it is a perfect code in the Cayley graph Cay(G,X). In this paper we present various families of examples of such codes, with X closed under conjugation and Y a subgroup, in symmetric groups, and also in special linear groups SL2(q). We also propose conjectures about the existence of some much wider families.

Burness TC, Liebeck MW, Shalev A, 2020, The length and depth of compact Lie groups, *Mathematische Zeitschrift*, Vol: 294, Pages: 1457-1476, ISSN: 0025-5874

Let G be a connected Lie group. An unrefinable chain of G is defined to be a chain of subgroups G=G0>G1>⋯>Gt=1 , where each Gi is a maximal connected subgroup of Gi−1 . In this paper, we introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups G. We obtain best possible bounds on the length of G in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on dimG′ in terms of the chain difference of G, which is its length minus its depth.

Burness T, Liebeck M, Shalev A, 2019, On the length and depth of finite groups, *Proceedings of the London Mathematical Society*, Vol: 119, Pages: 1464-1492, ISSN: 1460-244X

An unrefinable chain of a finite group is a chain of subgroups = 0> 1>⋯> =1 , where each is a maximal subgroup of −1 . The length (respectively, depth) of is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth 3. Here, we go much further by determining the finite groups of depth 3 and 4. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most 9, which extends earlier work of Janko and Harada from the 1960s, and we use this to describe the structure of arbitrary finite groups of small length. We also present a number‐theoretic result of Heath‐Brown, which implies that there are infinitely many non‐abelian simple groups of length at most 9.Finally, we study the chain difference of (namely the length minus the depth). We obtain results on groups with chain differences 1 and 2, including a complete classification of the simple groups with chain difference 2, extending earlier work of Brewster et al. We also derive a best possible lower bound on the chain ratio (the length divided by the depth) of simple groups, which yields an explicit linear bound on the length of / ( ) in terms of the chain difference of , where ( ) is the soluble radical of .

Bridson MR, Evans DM, Liebeck MW,
et al., 2019, Algorithms determining finite simple images of finitely presented groups, *Inventiones Mathematicae*, Vol: 218, Pages: 623-648, ISSN: 0020-9910

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, for any fixed untwisted Lie type X there is an algorithm that determines whether or not any given finitely presented group has simple images of the form X(q) for infinitely many q, and if there are finitely many, the algorithm determines them.

Isaacs IM, Liebeck M, Navarro G,
et al., 2019, Fields of values of odd-degree irreducible characters, *Advances in Mathematics*, Vol: 354, ISSN: 0001-8708

In this paper we clarify the quadratic irrationalities that can be admittedby an odd-degree complex irreducible character χ of an arbitrary finite group. WriteQ(χ) to denote the field generated over the rational numbers by the values of χ, andlet d > 1 be a square-free integer. We prove that if Q(χ) = Q(√d) then d ≡ 1 (mod4) and if Q(χ) = Q(√−d), then d ≡ 3 (mod 4). This follows from the main result ofthis paper: either i ∈ Q(χ) or Q(χ) ⊆ Q(exp(2πi/m)) for some odd integer m ≥ 1.

Liebeck MW, Praeger CE, Saxl J, 2019, The classification of 3/2-transitive permutation groups and 1/2-transitive linear groups, *Proceedings of the American Mathematical Society*, Vol: 147, Pages: 5023-5037, ISSN: 1088-6826

A linear group G ≤ GL(V ), where V is a finite vector space, is called 12-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the 12-transitive linear groups. As a consequence we complete the determination of the finite 32-transitive permutation groups – the transitive groups for which a point-stabilizerhas all its nontrivial orbits of the same size. We also determine the (k +12)-transitive groups for integers k ≥ 2.

Liebeck M, Shalev A, 2019, Girth, words and diameter, *Bulletin of the London Mathematical Society*, Vol: 51, Pages: 539-546, ISSN: 0024-6093

We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.

Liebeck M, Halasi Z, Maroti A, 2019, Base sizes of primitive groups: bounds with explicit constants, *Journal of Algebra*, Vol: 521, Pages: 16-43, ISSN: 0021-8693

We show that the minimal base size b(G) of a finite primitive permutationgroup G of degree n is at most 2(log|G|/logn)+ 24. This bound is asymptotically best possible since there exists a sequence of primitivepermutation groups G of degrees n such that b(G)= 2(log|G|/log n) − 2and b(G) is unbounded. As a corollary we show that a primitive permutation group of degree n that does not contain the alternatinggroup Alt(n) has a base of size at most max {√n,25}.

Liebeck MW, Guralnick R, Shalev A,
et al., 2019, Zero-one generation laws for finite simple groups, *Proceedings of the American Mathematical Society*, Vol: 147, Pages: 2331-2347, ISSN: 0002-9939

Let G be a simple algebraic group over the algebraic closure of Fp (p prime), and let G (q) denote a corresponding finite group of Lie type over Fq, where q is a power of p. Let X be an irreducible subvariety of Gr for some r≥2. We prove a zero-one law for the probability that G(q) is generated by a random r-tuple in X(q) =X ∩ G(q)r : the limit of this probability as q increases (through values of q for which X is stable under the Frobenius morphism defining G(q)) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs G(q) to be generated by an r-tuple in X(q) for two sufficiently large values of q. We also prove a version of this result where the underlying characteristic is allowed to vary. In our main application, we apply these results to the case where r = 2 and the irreducible subvariety X = C × D, a product of two conjugacy classes of elements of finite order in G. This leads to new results on random (2, 3)-generation of finite simple groups G(q) of exceptional Lie type: provided G(q) is not a Suzuki group, we show that the probability that a random involution and a random element of order 3 generate G(q) tends to 1 as q→∞. Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and PSp4(q)) are randomly (2, 3)-generated. Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie type.

Burness T, Liebeck MW, Shalev A, 2019, The length and depth of algebraic groups, *Mathematische Zeitschrift*, Vol: 291, Pages: 741-760, ISSN: 0025-5874

Let G be a connected algebraic group. An unrefinable chain of G is a chain of subgroups G=G0>G1>⋯>Gt=1 , where each Gi is a maximal connected subgroup of Gi−1 . We introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group G in terms of the dimension of its unipotent radical Ru(G) and the dimension of a Borel subgroup B of the reductive quotient G/Ru(G) . In particular, a simple algebraic group of rank r has length dimB+r , which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group G exceeds 12dimG . We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most 6 (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group G with soluble radical R(G), the dimension of G / R(G) is bounded above in terms of the chain difference of G.

Guralnick R, Liebeck MW, 2019, Permutation representations of nonsplit extensions involving alternating groups, *Israel Journal of Mathematics*, Vol: 229, Pages: 181-191, ISSN: 0021-2172

L. Babai has shown that a faithful permutation representation of a nonsplit extension of a group by an alternating group Ak must have degree at least k2(12−o(1)), and has asked how sharp this lower bound is. We prove that Babai’s bound is sharp (up to a constant factor), by showing that there are such nonsplit extensions that have faithful permutation representations of degree 32k(k−1). We also reprove Babai’s quadratic lower bound with the constant 1/2 improved to 1 (by completely different methods).

Lee M, Liebeck MW, 2018, Bases for quasisimple linear groups, *Algebra and Number Theory*, Vol: 12, Pages: 1537-1557, ISSN: 1937-0652

Let V be a vector space of dimension d over Fq, a finite field of q elements, and let G≤GL (V)∼=GLd(q) be a linear group. A basefor G is a set of vectors whose pointwise stabiliser in G is trivial. We prove that if G is a quasisimple group (i.e. Gis perfect and G/Z (G) is simple) acting irreducibly on V, then excluding two natural families, G has a base of size at most 6. The two families consist of alternating groups Alt m acting on the natural module of dimension d=m−1 orm−2, and classical groups with natural module of dimension d over subfields of Fq.

Bezrukavnikov R, Liebeck MW, Shalev A,
et al., 2018, Character bounds for finite groups of Lie type, *Acta Mathematica*, Vol: 221, Pages: 1-57, ISSN: 0001-5962

We establish new bounds on character values and character ratios for finite groups G of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form |χ(g)| ≤ cχ(1)αg , and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular we deduce that, if G is a classical group in dimension n, then, under some conditions on G and g ∈ G, the mixing time of the random walk on G with the conjugacy class of g as a generating

Guralnick R, Liebeck MW, O'Brien E,
et al., 2018, Surjective word maps and Burnside's p^a q^b theorem, *Inventiones Mathematicae*, Vol: 213, Pages: 589-695, ISSN: 0020-9910

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x,y)↦xNyN is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x,y,z)↦xNyNzN is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map (x,y)↦xNyN that depend on the number of prime factors of the integer N.

Burness TC, Liebeck MW, Shalev A, 2018, The depth of a finite simple group, *Proceedings of the American Mathematical Society*, Vol: 146, Pages: 2343-2358, ISSN: 0002-9939

We introduce the notion of the depth of a finite group G, defined as theminimal length of an unrefinable chain of subgroups from G to the trivial subgroup. Inthis paper we investigate the depth of (non-abelian) finite simple groups. We determinethe simple groups of minimal depth, and show, somewhat surprisingly, that alternatinggroups have bounded depth. We also establish general upper bounds on the depth ofsimple groups of Lie type, and study the relation between the depth and the much studiednotion of the length of simple groups. The proofs of our main theorems depend (amongother tools) on a deep number-theoretic result, namely, Helfgott’s recent solution of theternary Goldbach conjecture.

Liebeck MW, Burness TC, Shalev A, 2017, Generation of second maximal subgroups and the existence of special primes, *Forum of Mathematics, Sigma*, Vol: 5, ISSN: 2050-5094

Let G be a finite almost simple group. It is well known that G can be generated by three elements,and in previous work we showed that 6 generators suffice for all maximal subgroups of G. Inthis paper, we consider subgroups at the next level of the subgroup lattice—the so-called secondmaximal subgroups. We prove that with the possible exception of some families of rank 1 groupsof Lie type, the number of generators of every second maximal subgroup of G is bounded by anabsolute constant. We also show that such a bound holds without any exceptions if and only if thereare only finitely many primes r for which there is a prime power q such that (qr − 1)/(q − 1)is prime. The latter statement is a formidable open problem in Number Theory. Applications torandom generation and polynomial growth are also given.

Liebeck MW, Schul G, Shalev A, 2017, Rapid growth in finite simple groups, *Transactions of the American Mathematical Society*, Vol: 369, Pages: 2765-8779, ISSN: 1088-6850

We show that small normal subsets A of finite simple groups growvery rapidly – namely, |A2| ≥ |A|2− , where > 0 is arbitrarily small.Extensions, consequences, and a rapid growth result for simple algebraicgroups are also given.

Leemans D, Liebeck MW, 2017, Chiral polyhedra and finite simple groups, *Bulletin of the London Mathematical Society*, Vol: 49, Pages: 581-592, ISSN: 1469-2120

We prove that every finite non-abelian simple group acts asthe automorphism group of a chiral polyhedron, apart from the groupsP SL2(q),P SL3(q),P SU3(q) andA7.

Liebeck MW, Thomas AR, 2017, Finite subgroups of simple algebraic groups with irreducible centralizers, *Journal of Group Theory*, Vol: 20, Pages: 841-870, ISSN: 1433-5883

We determine all finite subgroups of simple algebraic groups that have irre-ducible centralizers – that is, centralizers whose connected component does notlie in a parabolic subgroup.

Liebeck MW, 2016, Character ratios for finite groups of Lie type, and applications, *Contemporary Mathematics*, Vol: 694, ISSN: 0271-4132

For a nite groupG, acharacter ratiois a complex number of the form (x) (1),wherex2Gand is an irreducible character ofG. Upper bounds for absolutevalues of character ratios, particularly for simple groups, have long been of interest,for various reasons; these include applications to covering numbers, mixing timesof random walks, and the study of word maps. In this article we shall survey someresults on character ratios for nite groups of Lie type, and their applications.Character ratios for alternating and symmetric groups have been studied in greatdepth also { see for example [32, 33] { culminating in the de nitive results andapplications to be found in [20]; but we shall not discuss these here.It is not hard to see the connections between character ratios and group struc-ture. Here are three well known, elementary results illustrating these connections.The rst two go back to Frobenius. Denote by Irr(G) the set of irreducible charac-ters ofG.

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