Petr Vojtechovsky
University of Denver, USA
Enumeration of nilpotent loops by means of cohomology.
Abstract: Loops are groupoids with neutral element in which the equations ax=b and ya=b have unique solutions x, y for given a, b. Informally, loops are "not necessarily associative groups". The concept of nilpotency in loops is defined analogously to that in groups. I will show how to count nilpotent loops of small orders up to isomorphism. After developing the correct cohomology theory, we will consider coboundaries and the action of the automorphism group on cocycles. If the resulting equivalence classes of cocycles coincide with isomorphism classes (a situation we call "separable"), the loops can be counted by purely linear algebraic methods, in some cases by hand. We will show how to deal with the inseparable situation, too. This is joint work with Daniel Daly.