I shall expect the audience to remember what is meant by a group, but not much else. Some groups are finite (i.e. have a finite number of elements), and some are not. I shall be talking about finite groups. The building blocks for all finite groups are the so-called simple groups. These are the finite groups that cannot be factorised, a bit like prime numbers in number theory. One of the major achievements of 20th century algebra was the classification of all the finite simple groups, achieved over a 50 year period by many mathematicians in several hundred journal articles covering thousands of pages.
A fundamental construction in group theory and other subjects is the commutator of two elements x and y. This is defined to be the element [x,y] = x^{-1}y^{-1}xy, and provides a measure of how far x and y are from commuting with each other. In 1951 Oystein Ore conjectured that every element of every non-commutative finite simple group can be expressed as the commutator of two other elements. Many people worked on Ore’s conjecture over the years, and it was finally proved in 2010 by Liebeck, O’Brien, Shalev and Tiep. It is a result I therefore modestly refer to now as Liebeck’s Lost Theorem.
In the talk I will try to introduce you to a few simple groups, explain why Ore’s conjecture is perhaps of some interest, and even give some indications about how it was proved.
Refreshments will be served at 4pm after the talk in the Huxley Common Room