1 - Preliminary results

Summary

I assume familiarity with material from a standard course on elementary algebra. A typical text for such a course is Herstein [He]. A few deeper algebraic results are also needed; they can be found for example in Lang [La]. Section 1 lists the elementary group theoretic results assumed and also contains a list of basic notation. Later sections in chapter 1 introduce some terminology and notation from a few other areas of algebra. Deeper algebraic results are introduced when they are needed.

The last section of chapter 1 contains a brief discussion of group representations. The term representation is used here in a more general sense than usual. Namely a representation of a group G will be understood to be a group homo-morphism of G into the group of automorphisms of an object X. Standard use of the term representation requires X to be a vector space.

Elementary group theory

Recall that a binary operation on a set G is a function from the set product G×G into G. Multiplicative notation will usually be used. Thus the image of a pair (x, y) under the binary operation will be written xy. The operation is associative if (xy)z = x(yz) for all x, y, z in G. The operation is commutative if xy = yx for all x, y in G.