Abstract:

What is the maximal size of an abelian subgroup of a group? What is the index of the centre? Such questions originate from Jordan and Baer and were later considered by Erdos and his co-workers. Recently many of these questions have been answered in a satisfactory way.

In the last two decades another way of measuring "how abelian a group is" has appeared. In particular it has been proved that every simple group of Lie-type is the set-theoretic product of 10000 abelian subgroups. The talk aims at giving a broad survey of these developments.