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On the uniform spread of almost simple linear groups
Full text linkLet G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial elements x(1),...,x(k) in G there exists y is an element of C such that G = <x(i), y > for all i. Further, the exact uniform spread of G, denoted by u(G), is the largest k such that G has the uniform spread k property. By a theorem of Breuer, Guralnick, and Kantor, u(G) > 1 for every finite simple group G. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if G = <PSLn(q),g > is almost simple, then u(G) > 1 (unless G is isomorphic to S_6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.</p
Fixed point spaces in actions of classical algebraic groups
No full textLet G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0, and let H be a maximal closed non-subspace subgroup of G. Given such a
pair (G, H), we obtain a close to best possible upper bound for the ratio dim(xG ? H) / =dim xG, where x ? G is a semisimple or unipotent element of prime order. We apply this result to the
study of fixed point spaces
Fixed point spaces in primitive actions of simple algebraic groups
No full textLet G be a simple algebraic group of adjoint type acting primitively on an algebraic variety ?. We study the dimensions of the subvarieties of fixed points of involutions in G. In particular, we obtain a close to best possible function f(h), where h is the Coxeter number of G, with the property that with the exception of a small finite number of cases, there exists an involution t in G such that the dimension of the fixed point space of t is at least f(h)dim?
Fixed point ratios in actions of finite classical groups, IV
Full text linkThis is the final paper in a series of four on fixed point ratios in non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and ? is a faithful transitive non-subspace G-set then either fpr(x) ~< |xG|-1/2 for all elements x?G of prime order, or (G,?) is one of a small number of known exceptions. In this paper we assume G? is either an almost simple irreducible subgroup in Aschbacher's ? collection, or a subgroup in a small additional set N which arises when G has socle Sp4(q)? (q even) or P?8+(q). This completes the proof of the main theorem
Irreducible almost simple subgroups of classical algebraic groups
No full textLet G be a simple classical algebraic group over an algebraically closed field K of characteristic p ≥ 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable p-restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V ) of this form, where H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension
On base sizes for actions of finite classical groups
No full textLet G be a finite almost simple classical group and let
? be a faithful primitive non-standard G-set. A base for G is a subset B C_ ? whose pointwise stabilizer is trivial; we write b(G) for the minimal size of a base for G. A well-known conjecture of Cameron and Kantor asserts that there exists an absolute constant c such that b(G) ? c for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) ? 4, or G = U6(2).2, G? = U4(3).22 and b(G) = 5.
The proof is probabilistic, using bounds on fixed point ratios
Base sizes for simple groups and a conjecture of Cameron
Full text linkLet G be a permutation group on a finite set ?. A base for G is a subset B C_ ? whose pointwise stabilizer in G is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ? if G is an almost simple group of exceptional Lie type and is a primitive faithful G-set. An important consequence
of this result, when combined with other recent work, is that b(G) ? 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios
Extremely primitive sporadic and alternating groups
No full textA non-regular primitive permutation group is said to be extremely primitive if a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In a recent paper, we classified the extremely primitive almost simple classical groups, and in this note we determine the examples with a sporadic or alternating socle. We obtain two infinite families for An (or Sn); they comprise the natural 2-primitive action of n points, plus the action on partitions of {1,?…?, n} into subsets of size n/2 (with n/2 odd). There are twenty examples for sporadic groups, including the rank 6 representation of Co2 on the cosets of McL
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