We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson’s group F. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the group has a wandering interval. In the latter case, the chain group admits a c...

Let G be a nite group and be a set of primes. Put d (G) = k (G)=jGj where k (G) is the number of conjugacy classes of -elements in G and jGj is the -part of the order of G. In this paper we initiate the study of this invariant by showing that if d (G) > 5=8 then G possesses an abelian Hall -subgroup, all Hall -subgroups of G are conjugate, and every -subgroup of G lies in some Hall -subgroup of...

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an ominimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R ·S where R is the solvable radical of G. We also prove that any semisimple subalgebra of t...

We solve the following problem: Let M be a 2g × 2g symplectic matrix of prime order with integer entries. Find a unique normal form for M , that is, a symplectic matrix whose entries are determined by its conjugacy invariants. This is equivalent to finding a unique normal form for the matrix representation of the conjugacy class of a prime order element of the mapping class group.

Let G be a classical group over an algebraically closed field of characteristic 2 and let C be an elliptic conjugacy class in the Weyl group. In a previous paper the first named author associated to C a unipotent conjugacy class Φ(C) of G. In this paper we show that Φ(C) can be characterized in terms of the closure relations between unipotent classes. Previously, the analogous result was known ...

Let G be a classical group over an algebraically closed field of characteristic 2 and let C be an elliptic conjugacy class in the Weyl group. In a previous paper the first named author associated to C a unipotent conjugacy class Φ(C) of G. In this paper we show that Φ(C) can be characterized in terms of the closure relations between unipotent classes. Previously, the analogous result was known ...