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On the uniform spread of almost simple linear groups
Get PDFLet 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
Get PDFThis 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
On irreducible subgroups of simple algebraic groups
Get PDFLet G be a simple algebraic group over an algebraically closed field K of characteristic p > 0, let H be a proper closed subgroup of G and let V be a nontrivial irreducible KG-module, which is p-restricted, tensor indecomposable and rational. Assume that the restriction of V to H is irreducible. In this paper, we study the triples (G, H, V ) of this form when G is a classical group and H is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples (G, H, V ) to the case where G is an orthogonal group, V is a spin module and H normalizes an orthogonal decomposition of the natural KG-module
Irreducible almost simple subgroups of classical algebraic groups
Get PDFLet 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 p-restricted irreducible tensor indecomposable 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 V ≠ W, W∗ and H is a disconnected almost simple positive-dimensional closed 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
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