Band 16, Heft 6

Abstract. The non-commuting graph of a finite group G is a highly symmetrical object (indeed, embeds in ), yet its complexity pales in comparison to that of G . Still, it is natural to seek conditions under which G can be reconstructed from . Surely some conditions are necessary, as is evidenced by the minuscule example . A conjecture made in [J. Algebra 298 (2006), 468–492], commonly referred to as the AAM Conjecture, proposes that the property of being a nonabelian simple group is sufficient. In [Sib. Math. J. 49 (2008), no. 6, 1138–1146], this conjecture is verified for all sporadic simple groups, while in [J. Algebra 357 (2012), 203–207], it is verified for the alternating groups. In this paper we verify it for the simple groups of Lie type, thereby completing the proof of the conjecture.

Abstract. An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element s in a dual group such that . We prove that there is a bijection between, on the one hand, the set of quadratic unipotent characters of or for all and, on the other hand, the set of quadratic unipotent characters of for all . We then extend this correspondence to ℓ-blocks for certain ℓ not dividing q .

Abstract. Using the classical results of G. E. Wall on the parametrization and sizes of (conjugacy) classes of finite classical groups, we present some gap results for the class sizes of the general linear groups and general unitary groups as well as their variations. In particular, we identify the classes in of size up to and classes in of size up to . We then apply these gap results to obtain some bounds and limits concerning the zeta-type function encoding the conjugacy class sizes of these groups.

Abstract. We introduce three families of characteristic subgroups that refine the traditional verbal subgroup filters, such as the lower central series, to an arbitrary length. We prove that a positive logarithmic proportion of finite p -groups admit at least five such proper nontrivial characteristic subgroups whereas verbal and marginal methods explain only one. The placement of these subgroups in the lattice of subgroups is naturally recorded by a filter over an arbitrary commutative monoid M and induces an M -graded Lie ring. These Lie rings permit an efficient specialization of the nilpotent quotient algorithm to construct automorphisms and decide isomorphism of finite p -groups. As a demonstration, we identify some large families of p -groups that are worst-case examples for the traditional nilpotent quotient algorithm but run in polynomial time when using our new filters.

Abstract. Erratum to http://dx.doi.org/10.1515/jgt-2013-0026.

Abstract. In this article we extend the result of Guralnick, Kantor, Kassabov and Lubotzky to the affine Kac–Moody groups: we show that there exists a constant such that every affine Kac–Moody group defined over a finite field , (with the exception of and ), has a presentation σ with . We then derive the consequences of this result for the 2-spherical Kac–Moody groups.

Abstract. In [J. Lie Theory 13 (2003), no. 2, 383–385], the authors introduced a framework to prove that a large class of HNN extensions have the Haagerup property, the main motivation being Baumslag–Solitar groups. Using this framework and new tools on locally compact groups developed in [“Amenable hyperbolic groups”, preprint (2012)], we are able to obtain quantitative results on embeddings into Lebesgue spaces for a large class of HNN extensions.

Abstract. A subgroup  H of an Abelian group  G is said to be fully inert if the quotient is finite for every endomorphism ϕ of G . Clearly, this is a common generalization of the notions of fully invariant, finite and finite-index subgroups. We investigate the fully inert subgroups of divisible Abelian groups, and in particular, those Abelian groups that are fully inert in their divisible hull, called inert groups. We prove that the inert torsion-free groups coincide with the completely decomposable homogeneous groups of finite rank and we give a complete description of the inert groups in the general case. This yields a characterization of the fully inert subgroups of divisible Abelian groups.

Abstract. Let be the n -dimensional quaternionic hyperbolic space. The group acts as the isometry group of . We analyze when two isometries of commute. We apply this analysis to determine the conjugacy classes of centralizers or the z -classes in . Furthermore, we count the conjugacy classes of centralizers. In Appendix A, we show that our methods can be used to obtain the centralizers up to conjugacy in real and complex hyperbolic geometries as well. This provides a unified approach to the determination of the conjugacy classes of centralizers in hyperbolic geometries.

Abstract. We consider diagram groups as defined by Guba and Sapir [Mem. Amer. Math. Soc. 130 (1997)]. A diagram group G acts on the associated cube complex K by isometries. It is known that if a cube complex L is of a finite dimension, then every isometry g of L is semi-simple: is attained. It was conjectured by Farley that in the case of a diagram group G the action of G on the associated cube complex K is by semisimple isometries also when K has an infinite dimension. In this paper we give a counter-example to Farley's conjecture by showing that R. Thompson's group F , considered as a diagram group, has some elements which act as parabolic (not semi-simple) isometries on the associated cube complex.