We have have encountered several finite groups in the previous chapter: the cyclic group Zn, the symmetric group Sn, and its subgroup the alternating group An. The properties of these groups will be elaborated in this chapter. We will also discuss three other groups obtained by incorporating ‘reflections’ into Zn, the dihedral group Dn, the generalized quaterion group Qn, and the dicyclic group...

This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In §2 we collect some questions that motivated the lectures and this article, and in §3 discu...

We have already met covering spaces in Chapter 2, where our discussion was geometric and basically limited to definitions and examples. In this chapter, we return to this topic from a more algebraic point of view, which will allow us to produce numerous examples coming from group actions and to classify all covering spaces with given base (provided the latter is “nice” enough). The main tools w...

In this note we develop a theory of formal schemes and groups over arbitrary commutative rings which coincides with that of [5] if the base ring is a field, and generalizes that of [2]. We assume always our base ring is discrete and treat a formal scheme (resp. group) G, with two principal tools: A topology on the affine algebra (9(G) allows us to form its continuous linear dual B(G), the coalg...

These notes summarize four expository lectures delivered at the Advanced School of the ICTS Program Groups, Geometry and Dynamics, December, 2012, Almora, India. The target audience was a group of students at or near the end of a traditional undergraduate math major. My purpose was to expose the types of discrete groups that arise in connection with Riemann surfaces. I have not hesitated to sho...