134 results found
Halasi Z, Liebeck MW, Maroti A, 2019, Base sizes of primitive groups: Bounds with explicit constants, JOURNAL OF ALGEBRA, Vol: 521, Pages: 16-43, ISSN: 0021-8693
Liebeck MW, Guralnick R, Shalev A, et al., Zero-one generation laws for finite simple groups, Proceedings of the American Mathematical Society, ISSN: 0002-9939
LetGbe a simple algebraic group over the algebraic closure ofFp(pprime), and letG(q) denote a corresponding finite group of Lie type overFq, whereqis a power ofp. LetXbean irreducible subvariety ofGrfor somer≥2. We prove a zero-one law for the probability thatG(q) is generated by a randomr-tuple inX(q) =X∩G(q)r: the limit of this probability asqincreases (through values ofqfor whichXis stable under the Frobenius morphism definingG(q))is either 1 or 0. Indeed, to ensure that this limit is 1, one only needsG(q) to be generated by anr-tuple inX(q) for two sufficiently large values ofq. We also prove a version of this result wherethe underlying characteristic is allowed to vary.In our main application, we apply these results to the case wherer= 2 and the irreduciblesubvarietyX=C×D, a product of two conjugacy classes of elements of finite order inG. Thisleads to new results on random (2,3)-generation of finite simple groupsG(q) of exceptional Lietype: providedG(q) is not a Suzuki group, we show that the probability that a random involutionand a random element of order 3 generateG(q) tends to 1 asq→∞. Combining this with previousresults for classical groups, this shows that finite simple groups (apart from Suzuki groups andPSp4(q)) are randomly (2,3)-generated.Our tools include algebraic geometry, representation theory of algebraic groups, and charactertheory of finite groups of Lie type.
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
Burness T, Liebeck MW, Shalev A, The length and depth of algebraic groups, Mathematische Zeitschrift, ISSN: 0025-5874
LetGbe a connected algebraic group. An unrefinable chain ofGis a chainof subgroupsG=G0> G1>···> Gt= 1, where eachGiis a maximal connectedsubgroup ofGi−1. We introduce the notion of the length (respectively, depth) ofG,defined as the maximal (respectively, minimal) length of such a chain. Working over analgebraically closed field, we calculate the length of a connected groupGin terms of thedimension of its unipotent radicalRu(G) and the dimension of a Borel subgroupBofthe reductive quotientG/Ru(G). In particular, a simple algebraic group of rankrhaslength dimB+r, which gives a natural extension of a theorem of Solomon and Turull onfinite quasisimple groups of Lie type. We then deduce that the length of any connectedalgebraic groupGexceeds12dimG.We also study the depth of simple algebraic groups. In characteristic zero, we showthat the depth of such a group is at most 6 (this bound is sharp). In the positivecharacteristic setting, we calculate the exact depth of each exceptional algebraic groupand we prove that the depth of a classical group (over a fixed algebraically closed fieldof positive characteristic) tends to infinity with the rank of the group.Finally we study the chain difference of an algebraic group, which is the differencebetween its length and its depth. In particular we prove that, for any connected algebraicgroupGwith soluble radicalR(G), the dimension ofG/R(G) is bounded above in termsof the chain difference ofG.
Guralnick R, Liebeck MW, Permutation representations of nonsplit extensions involving alternating groups, Israel Journal of Mathematics, ISSN: 0021-2172
L. Babai has shown that a faithful permutation represen-tation of a nonsplit extension of a group by an alternating groupAkmust have degree at leastk2(12−o(1)), and has asked how sharp thislower bound is. We prove that Babai’s bound is sharp (up to a con-stant factor), by showing that there are such nonsplit extensions thathave faithful permutation representations of degree32k(k−1). We alsoreprove Babai’s quadratic lower bound with the constant12improved to1 (by completely different methods).
Lee M, Liebeck MW, 2018, Bases for quasisimple linear groups, ALGEBRA & NUMBER THEORY, Vol: 12, Pages: 1537-1557, ISSN: 1937-0652
Liebeck MW, Schul G, Shalev A, 2017, RAPID GROWTH IN FINITE SIMPLE GROUPS, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 369, Pages: 8765-8779, ISSN: 0002-9947
Burness TC, Liebeck MW, Shalev A, 2017, GENERATION OF SECOND MAXIMAL SUBGROUPS AND THE EXISTENCE OF SPECIAL PRIMES, FORUM OF MATHEMATICS SIGMA, Vol: 5, ISSN: 2050-5094
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
Leemans D, Liebeck MW, 2017, Chiral polyhedra and finite simple groups, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Vol: 49, Pages: 581-592, ISSN: 0024-6093
Gonshaw S, Liebeck MW, O'Brien EA, 2017, Unipotent class representatives for finite classical groups, JOURNAL OF GROUP THEORY, Vol: 20, Pages: 505-525, ISSN: 1433-5883
Liebeck MW, 2017, Character ratios for finite groups of Lie type, and applications, FINITE SIMPLE GROUPS: THIRTY YEARS OF THE ATLAS AND BEYOND, Vol: 694, Pages: 193-208, ISSN: 0271-4132
Liebeck MW, O'Brien EA, 2016, RECOGNITION OF FINITE EXCEPTIONAL GROUPS OF LIE TYPE, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 368, Pages: 6189-6226, ISSN: 0002-9947
Liebeck MW, Praeger CE, Saxl J, The classification of 3/2-transitive permutation groups and 1/2-transitive linear groups, Proceedings of the American Mathematical Society, ISSN: 1088-6826
A linear group G ≤ GL(V ), where V is a finite vector space, is called 12-transitive if allthe G-orbits on the set of nonzero vectors have the same size. We complete the classificationof all the 12-transitive linear groups. As a consequence we complete the determination of thefinite 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 groupsfor integers k ≥ 2.
Liebeck MW, Seitz GM, Testerman DM, 2015, DISTINGUISHED UNIPOTENT ELEMENTS AND MULTIPLICITY-FREE SUBGROUPS OF SIMPLE ALGEBRAIC GROUPS, PACIFIC JOURNAL OF MATHEMATICS, Vol: 279, Pages: 357-382, ISSN: 0030-8730
Liebeck MW, 2015, On products of involutions in finite groups of Lie type in even characteristic, JOURNAL OF ALGEBRA, Vol: 421, Pages: 431-437, ISSN: 0021-8693
Liebeck MW, Shalev A, 2015, On fixed points of elements in primitive permutation groups, JOURNAL OF ALGEBRA, Vol: 421, Pages: 438-459, ISSN: 0021-8693
Liebeck MW, Shalev A, 2014, POWER SETS AND SOLUBLE SUBGROUPS, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 142, Pages: 3757-3760, ISSN: 0002-9939
Lawther R, Liebeck MW, Seitz GM, 2014, Outer unipotent classes in automorphism groups of simple algebraic groups, PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, Vol: 109, Pages: 553-595, ISSN: 0024-6115
Liebeck MW, Shalev A, 2014, Bases of primitive linear groups II, JOURNAL OF ALGEBRA, Vol: 403, Pages: 223-228, ISSN: 0021-8693
Liebeck MW, Shalev A, 2013, POWERS IN FINITE GROUPS AND A CRITERION FOR SOLUBILITY, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 141, Pages: 4179-4189, ISSN: 0002-9939
Burness TC, Liebeck MW, Shalev A, 2013, Generation and random generation: From simple groups to maximal subgroups, ADVANCES IN MATHEMATICS, Vol: 248, Pages: 59-95, ISSN: 0001-8708
Jambor S, Liebeck MW, O'Brien EA, 2013, Some word maps that are non-surjective on infinitely many finite simple groups, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Vol: 45, Pages: 907-910, ISSN: 0024-6093
Bamberg J, Giudici M, Liebeck MW, et al., 2013, THE CLASSIFICATION OF ALMOST SIMPLE 3/2-TRANSITIVE GROUPS, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 365, Pages: 4257-4311, ISSN: 0002-9947
Liebeck MW, 2013, Probabilistic and Asymptotic Aspects of Finite Simple Groups, PROBABILISTIC GROUP THEORY, COMBINATORICS, AND COMPUTING: LECTURES FROM THE FIFTH DE BRUN WORKSHOP, Editors: Detinko, Flannery, OBrien, Publisher: SPRINGER-VERLAG LONDON LTD, Pages: 1-34, ISBN: 978-1-4471-4813-5
Liebeck MW, Nikolov N, Shalev A, 2012, Product decompositions in finite simple groups, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Vol: 44, Pages: 469-472, ISSN: 0024-6093
Liebeck MW, Segal D, Shalev A, 2012, The density of representation degrees, JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, Vol: 14, Pages: 1519-1537, ISSN: 1435-9855
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