Let G be an algebraic group of type G2 over a field k of characteristic 6= 2, 3. In this paper we calculate centralizers of semisimple elements in anisotropic G2. Using these, we show explicitly that there are six conjugacy classes of centralizers in the compact real form of G2. AMS Subject Classification: 20G20, 17A75.

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a “matrix problem”. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a ...

The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ⊂ GL(n,C) or cj ⊂ gl(n,C) so that there exist irreducible (resp. with trivial centralizer) (p + 1)-tuples of matrices Mj ∈ Cj or Aj ∈ cj satisfying the equality M1 . . .Mp+1 = I or A1+ . . .+Ap+1 = 0. The matrices Mj and Aj are inter...

Conjugacy classes of outer automorphisms of order 3 of simple algebraic groups of classical type D4 are classified over arbitrary fields. There are two main types of conjugacy classes. For one type the fixed algebraic groups are simple of type G2; for the other type they are simple of type A2 when the characteristic is different from 3 and are not smooth when the characteristic is 3. A large pa...

In this survey paper we show how character methods can be used to solve a wide range of seemingly unrelated problems. These include commutators, powers of conjugacy classes and related random walks, as well as word maps and Waring type problems. In particular we describe recent progress made on conjectures of Ore, of Thompson, and of Lulov and Pak. New open problems and conjectures are also sta...

Of course, in that problem we have to take into account that the class sizes impose restrictions on the group structure. E.g. if the sizes are {1, p}, then the nilpotency class has to be 2. More precisely: the class sizes of a p-group G are {1, p} iff |G′| = p (Knoche; see also Theorem 3 below). But we can ask, e.g., if, given any set S ≠ {1, p} of p-powers, does there exist a group of class 3 ...