I have just finished a crash course on group theory at Universidade de Lisboa. The notes are here.
From the preface:
On a visit to Universidade de Lisboa in November 2016, I was asked to give a “crash course” in group theory. The only specifications were that the course should cover both finite and infinite groups and should be accessible to students.
This is a tall order. I have tried to meet it by starting at the beginning, moving fairly fast, omitting many proofs (this means leaving many proofs to the reader). But I hope the result is still of some use, so I am making the notes of the course available.
There is a clear focus in the chapter on finite groups: we want to be able to describe them all. The Jordan–Hölder theorem reduces the problem to describing the finite simple groups and how general groups are built out of simple groups: the Classification of Finite Simple Groups solves the first part; a discussion of groups of prime power order shows that we cannot expect a nice solution to the second.
For infinite groups, such a focus is much more difficult to obtain. There is no general theory of infinite groups, and group theorists have imposed various finiteness conditions on their groups. I discuss, somewhat in the manner of a tourist guide, free groups, presentations of groups, periodic and locally finite groups, residually finite and profinite groups, and my own interest, oligomorphic permutation groups.
I thank the students for their interest and their questions.