139 results found
Liebeck M, Burness T, Shalev A, On the length and depth of finite groups, Proceedings of the London Mathematical Society, ISSN: 1460-244X
Burness TC, Liebeck MW, Shalev A, The length and depth of compact Lie groups, Mathematische Zeitschrift, ISSN: 0025-5874
Liebeck M, Bridson M, Evans D, et al., Algorithms determining finite simple images of finitely presented groups, Inventiones Mathematicae, ISSN: 0020-9910
Benkart G, Diaconis P, Liebeck M, et al., Tensor product Markov chains, Journal of Algebra, 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.
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 M, Shalev A, Girth, words and diameter, Bulletin of the London Mathematical Society, 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.
Burness TC, Liebeck MW, Shalev A, 2019, The length and depth of algebraic groups, MATHEMATISCHE ZEITSCHRIFT, Vol: 291, Pages: 741-760, ISSN: 0025-5874
Guralnick RM, Liebeck MW, 2019, Permutation representations of nonsplit extensions involving alternating groups, ISRAEL JOURNAL OF MATHEMATICS, Vol: 229, Pages: 181-191, ISSN: 0021-2172
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
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
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