Summary of Chapter 8
Overview of sections
Summary of contents
Chapter 8 is concerned with permutation groups. It deals with a slightly more general setting, namely that of a permutation representation: a morphism of groups f : G -> Sym(X), where X is any set. In practice, however, we frequently restrict ourselves to a finite set and label its elements from 1 to n, so that
We discuss notions such as orbits and transitivity and order. In particular, we describe an algorithm for determining the order of a permutation group given by a set of generating permutations.
Groups often appear as automorphism groups of other structures, like graphs and geometries, or algebraic structures, like rings and fields. We consider various examples of such automorphism groups.
We also consider normal subgroups of groups and quotients. We show that each normal subgroup is the kernel of a group morphism onto the quotient group.
We end the chapter with a classification of all groups of order at most 11 (or even 15?).