Conjugacy classes in finite groups

In the first part of this note, we give new proofs of known results regarding the class number of finite groups, adding a few related results. In the second part, we improve a result of Ito concerning a special class ofp-groups.     

Conjugacy classes in sporadic simple groups

We obtain efficient programs for obtaining representatives of all the conjugacy classes of all the sporadic simple groups except the Monster and the Baby Monster.     

Conjugacy classes in finite solvable groups

In this note, we obtain the number of conjugacy classes in a finite solvable group as a function of any tuple of the composition factors ofG. Using this relation, we give a new elementary proof of one of Mann's results for solvable groups, without using character theory, and we improve this result for some classes of groups.     

Conjugacy classes of groups of bundle automorphisms

Let B be a space which admits a numerable covering {U : } with the property that every principal G-bundle over B is locally trivial with respect to the covering {U }; let G(p) be the space of all equivariant automorphisms of p. In this setting the groups G(p) can be viewed as subgroups of some common group which depends only on the covering {U } and G. We classify these groups G(p) up to conjugacy. In certain situations this leads to a characterization of the isomorphism classes of the groupsG(p).     

Conjugacy classes of symmetries in orthogonal groups

The number of conjugacy classes of symmetries in the integral orthogonal group of an indefinite -lattice is determined. The results are applied to the extended Bianchi groups.

     

Strongly dense free subgroups of semisimple algebraic groups

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type     

Conjugacy classes of derangements in finite transitive groups

Let G be a permutation group acting transitively on a finite set Ω. We classify all such (G, Ω) when G contains a single conjugacy class of derangements. This was done under the assumption that G acts primitively by Burness and Tong-Viet. It turns out that there are no imprimitive examples. We also discuss some results on the proportion of conjugacy classes which consist of derangements.     

Conjugacy classes in Weyl groups and q-W algebras

We define noncommutative deformations of algebras of regular functions on certain transversal slices to the set of conjugacy classes in an algebraic group G which play the role of Slodowy slices in algebraic group theory. The algebras called q-W algebras are labeled by (conjugacy classes of) elements s of the Weyl group of G. The algebra is a quantization of a Poisson structure defined on the corresponding transversal slice in G with the help of Poisson reduction of a Poisson bracket associated to a Poisson–Lie group G^? dual to a quasitriangular Poisson–Lie group. We also define a quantum group counterpart of the category of generalized Gelfand–Graev representations and establish an equivalence between this category and the category of representations of the corresponding q-W algebra. The algebras can be regarded as quantum group counterparts of W-algebras. However, in general they are not deformations of the usual W-algebras.     

Irreducible representations of semisimple algebraic groups and supersolvable lattices

Let Λ be the cross section lattice of an irreducible representation of a semisimple algebraic group. Certain combinatorial properties of Λ are studied. Supersolvable Λ?s are determined in terms of Dynkin diagrams. The characteristic polynomial of a supersolvable Λ is computed.     

Remarks on the wonderful compactification of semisimple algebraic groups

We prove that ifG is a semisimple algebraic group of adjoint type over the field of complex numbers,H is the subgroup of all fixed points of an involution σ ofG that is induced by an involution σ of the simply connected coveringĜ ofG, then the wonderful compactification of the homogeneous spaceG/H is isomorphic to the G.I.T quotientG ^ss (L)//H of the wonderful compactificationG ofG for a suitable choice of a line bundleL onG. We also prove a functorial property of the wonderful compactifications of semisimple algebraic groups of adjoint type.     

Conjugacy classes in finite permutation groups via homomorphic images

The lifting of results from factor groups to the full group is a standard technique for solvable groups. This paper shows how to utilize this approach in the case of non-solvable normal subgroups to compute the conjugacy classes of a finite group.

     

On the Grothendieck group of simply connected semisimple algebraic groups

The K₀ groups of simply connected semisimple algebraic groups are calculated. The triviality of the Chow groups CH¹ and CH² of such groups is obtained as a consequence. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 223–235.     

R-equivalence and rationality problem for semisimple adjoint classical algebraic groups

The group of R-equivalence classes for all adjoint semisimple classical algebraic groups is computed. Examples of stably non-rational adjoint simple groups of type D _n ,n≥3, are presented. The complete stable birational classification of adjoint simple groups of rank 3 is given. This paper was written while the author was on sabbatical leave at the Université catholique de Louvain. Typeset byA _M S-T_EX