## University degli Studi di Milano-Bicocca

In the first lecture of this course we review the O'Nan-Scott-Aschbacher
theorem in a form tailored to our needs. This theorem, combined with the
Classification of Finite Simple Groups, is a powerful tool for reducing
a problem concerning finite primitive groups to a case-by-case analysis.
Most of the time, this analysis is easier than the general problem;
moreover, this analysis sometimes offers some rather intriguing questions
within some special classes of finite groups
(for instance, almost simple groups).

In the remaining three lectures we give two applications of the O'Nan-Scott
theorem, one within the theory of finite primitive groups, and the other in
the theory of groups acting on finite graphs.

In the first application, we show that, except for a few examples, every
permutation of a finite primitive group has a cycle of length equal to
its order. This should offer a concrete example of how in practise a
problem on finite primitive groups can be solved with the
O'Nan-Scott-Aschbacher classification.

The second application is related to the enumeration of Cayley graphs and
we give an asymptotic formula for the number of Cayley graphs, up to
isomorphism, of an abelian group. We show how this problem reduces to
a problem on primitive groups, and then we use the O'Nan-Scott theorem
for solving the latter problem.