An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G = PSL(2, q). 2000 Mathematics subject classification: 20B25, 05C25.

We show that there exist a finite number of Cartan subgroups up to conjugacy in SL2(Qp) and we describe all of them. We show that the Cartan subgroup consisting of all diagonal matrices is generous and it is the only one up to conjugacy.

THEOREM. Let G be a finite group and s an involution in G, such that CG(S)~CGQ(SO)Assume G^Co(s)0(G) (0(G) is the maximal normal odd order subgroup of G). Then Gc^Go. In particular, G has the following properties : (i) G has order 2 • 3 • 5 • 7 • 11 • 23, and is simple. (ii) G has two conjugacy classes of involutions. One class is represented by the involution s. A representative t of the secon...

* Supported by a grant from the Graduiertenkolleg " Analyse und Konstruktion in der Mathematik " , RWTH Aachen.

Let G be a quasi-split connected reductive group over a local field of characteristic 0, and fix a regular nilpotent element in the Lie algebra g of G. A theorem of Kostant then provides a canonical conjugacy class within each stable conjugacy class of regular semisimple elements in g. Normalized transfer factors take the value 1 on these canonical conjugacy classes.

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0, 1 or −1 on C. In this paper we classify the sign conjugacy classes of the symmetric groups and thereby verify a conjecture of Olsson.

In this paper we introduce the graph Γsc(G) associated with a group G, called solvable conjugacy class (abbreviated as SCC-graph), whose vertices are nontrivial classes of G and two distinct C,D adjacent if there exist x∈C y∈D such that 〈x,y〉 is solvable. We discuss connectivity, girth, clique number, several other properties SCC-graph. One our results asserts only finitely many finite groups S...

Let G be a connected reductive linear algebraic group defined over an field k of characteristic not 2, θ ∈ Aut(G) an involutional k-automorphism of G and K = Gθ = {g ∈ G | θ(g) = g} the set of fixed points of θ. Denote the set of k-rational points of G by Gk. In this paper we shall classify the K-conjugacy classes of θ-stable maximal tori of G. This is shown to be independent of the characteris...