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  • Journal article
    Evans DM, Ghadernezhad Z, Tent K, 2016,

    Simplicity of the automorphism groups of some Hrushovski constructions

    , Annals of Pure and Applied Logic, Vol: 167, Pages: 22-48, ISSN: 1873-2461

    We show that the automorphism groups of certain countable structures obtained using the Hrushovski amalgamation method are simple groups. The structures we consider are the ‘uncollapsed’ structures of infinite Morley rank obtained by the ab initio construction and the (unstable) ℵ0-categorical pseudoplanes. The simplicity of the automorphism groups of these follows from results which generalize work of Lascar and of Tent and Ziegler.

  • Journal article
    Evans DM, Tsankov T, 2016,

    Free actions of free groups on countable structures and property (T)

    , Fundamenta Mathematicae, Vol: 232, Pages: 49-63, ISSN: 1730-6329

    We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ω-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a non-abelian free subgroup of G acting freely in all infinite transitive permutation representations of G.

  • Journal article
    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

    For G a simple algebraic group over an algebraically closed field of characteristic 0, we determine the irreducible representations ρ:G→I(V), where I(V) denotes one of the classical groups SL(V), Sp(V), SO(V), such that ρ sends some distinguished unipotent element of G to a distinguished element of I(V). We also settle a base case of the general problem of determining when the restriction of ρ to a simple subgroup of G is multiplicity-free.

  • Journal article
    Liebeck MW, O'Brien EA, 2015,

    Recognition of finite exceptional groups of Lie type

    , Transactions of the American Mathematical Society, Vol: 368, Pages: 6189-6226, ISSN: 1088-6850

    Let q be a prime power and let G be an absolutely irreducible subgroup ofGLd(F), where F is a finite field of the same characteristic as Fq, the field of q elements. Assume that G ∼= G(q), a quasisimple group of exceptional Lie type over Fq which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from G to the standard copy of G(q). If G 6∼=3D4(q) with q even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.

  • Journal article
    Franchi C, Ivanov AA, Mainardis M, 2015,

    The 2A Majorana representations of the Harada-Norton group.

    , ARS Mathematica Contemporanea, Vol: 11, ISSN: 1855-3966

    We show that all 2A-Majorana representations of the Harada-Norton groupF5have the same shape. IfRis such a representation, wedetermine, using the theory of association schemes, the dimension and theirreducible constituents of the linear spanUof the Majorana axes. Finally, weprove that, ifRis based on the (unique) embedding ofF5in the Monster,Uis closed under the algebra product.

  • Journal article
    Giudici M, Liebeck MW, Praeger CE, Saxl J, Pham HTet al., 2015,

    Arithmetic results on orbits of linear groups

    , Transactions of the American Mathematical Society, Vol: 368, Pages: 2415-2467, ISSN: 1088-6850
  • Journal article
    Amato D, Evans DM, 2015,

    Infinite primitive and distance transitive directed graphs of finite out-valency

    , Journal of Combinatorial Theory Series B, Vol: 114, Pages: 33-50, ISSN: 1096-0902

    We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.

  • Journal article
    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
  • Journal article
    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
  • Journal article
    Liebeck MW, Shalev A, 2014,

    POWER SETS AND SOLUBLE SUBGROUPS

    , PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 142, Pages: 3757-3760, ISSN: 0002-9939

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