Let M be an o–minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n–dimensional group definable in M. We show the following: the o–minimal fundamental group of G is isomorphic to Z; for each k > 0, the k–torsion subgroup of G is isomorphic to (Z/kZ), and the o–minimal cohomology algebra over Q of G is isomorphic to the exterior algebra over Q with ...

In this paper we review the theory of compact quantum groups.

We investigate the class of N -determined p-compact groups and the class of p-compact groups with N -determined automorphisms. A p-compact group is said to be N -determined if it is determined up to isomorphism by the normalizer of a maximal torus. The automorphisms of a p-compact group are said to be N -determined if they are determined by their restrictions to this maximal torus normalizer.

Compact groups of galaxies have posed a number of challenging questions. Intensive observational and theoretical studies are now providing answers to many of these, and at the same time, are revealing unexpected new clues about the nature and role of these systems. Most compact groups contain a high fraction of galaxies having morphological or kinematical peculiarities, nuclear radio and infrar...

The first half of the paper presents the basic definitions and results necessary for investigating Lie groups. The primary examples come from the matrix groups. The second half deals with representation theory of groups, particularly compact groups. The end result is the Peter-Weyl theorem.