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Claude Marion
May 30, 2010

### Abstract

This paper is concerned with the ( p 1 , p 2 , p 3 )-generation of finite groups of Lie type, where we say that a group is ( p 1 , p 2 , p 3 )-generated if it is generated by two elements of orders p 1 , p 2 having product of order p 3 . Given a triple ( p 1 , p 2 , p 3 ) of primes, we say that ( p 1 , p 2 , p 3 ) is rigid for a simple algebraic group G if the sum of the dimensions of the subvarieties of elements of orders dividing p 1 , p 2 , p 3 in G is equal to 2 dim G . We conjecture that if ( p 1 , p 2 , p 3 ) is a rigid triple for G then given a prime p , there are only finitely many positive integers r such that the finite group G ( p r ) is a ( p 1 , p 2 , p 3 )-group. We prove that the conjecture holds in many cases. Finally, we classify the rigid triples for simple algebraic groups. The conjecture together with this classification puts into context many results on Hurwitz (2, 3, 7)-generation in the literature, and motivates a new study of the ( p 1 , p 2 , p 3 )-generation problem for certain finite groups of Lie type of low rank.

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M. A. Pellegrini, M. C. Tamburini
April 23, 2010

### Abstract

We prove that the universal covering of an alternating group Alt( n ) which is Hurwitz is still Hurwitz, with 31 exceptions, 30 of which are detectable by the genus formula.

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W. Ethan Duckworth
April 23, 2010

### Abstract

Let G be an exceptional algebraic group, X a maximal rank reductive subgroup and P a parabolic subgroup. This paper classifies when X \ G / P is finite. Finiteness is proven using a reduction to finite groups and character theory. Infiniteness is proven using a dimension criterion that involves root systems.

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Ellen Henke
April 23, 2010

### Abstract

We show that, given a saturated fusion system, it is, under certain conditions, possible to identify SL 2 ( q ) acting on a natural module inside the normalizer of an essential subgroup. In particular, this is the case if the fusion system is non-constrained and has only one conjugacy class of essential subgroups.

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Dietrich Burde, Karel Dekimpe, Kim Vercammen
April 23, 2010

### Abstract

An LR-structure on a Lie algebra π€ is a bilinear product on π€, satisfying certain commutativity relations, and which is compatible with the Lie product. LR-structures arise in the study of simply transitive affine actions on Lie groups. In particular one is interested in the question which Lie algebras admit a complete LR-structure. In this paper we show that a Lie algebra admits a complete LR-structure if and only if it admits any LR-structure.

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J. Z. GonΓ§alves, D. S. Passman
April 23, 2010

### Abstract

If * : G β G is an involution on the finite group G , then * extends to an involution on the integral group ring β€[ G ]. In this paper, we consider whether bicyclic units u β β€[ G ] exist with the property that the group γ u, u *γ generated by u and u * is free on the two generators. If this occurs, we say that ( u, u *) is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. One positive result here is that if G is a nonabelian group with all Sylow subgroups abelian, then for any involution *, β€[ G ] contains a free bicyclic pair.

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Alessandro Martinelli
February 8, 2010

### Abstract

We prove that a residually nilpotent group with all subgroups subnormal is hypercentral and we bound the hypercentral length of a hypercentral group with all subgroups subnormal.

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Liad Fireman
April 23, 2010

### Abstract

Let S be a finite simple group, and the class of poly- S groups, that is, finite groups with all composition factors isomorphic to S . A pro- S group is defined to be an inverse limit of poly- S groups. If S = C p , the finite cyclic group of order p , we get the familiar pro- p groups. We study the case when S is non-abelian, and particularly the structure of free and projective pro- S groups and their subgroups. We show that pro- S groups have a rich structure, and that the categories of pro- p and pro- S groups are very different.

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Ian J. Leary, Brita E. A. Nucinkis
February 8, 2010

### Abstract

Let π denote the class of finite groups, and let π denote the subclass consisting of groups of prime-power order. We study group actions on topological spaces in which all stabilizers lie either in π or in π. We compare the classifying spaces for actions with stabilizers in π and π, the Kropholler hierarchies built on π and π, and group cohomology relative to π and to π. In terms of standard notation, we show that π β h 1 π β h 1 π, with all inclusions proper; that h π = h π; that π H *( G ; β); = π H *( G ; β); and that E π G is finite-dimensional if and only if E π G is finite-dimensional and every finite subgroup of G is in π.