Chapter 0 Permutation Actions This chapter is a very brief introduction to permutation group actions, pre- senting only topics that will re-appear in later chapters. For more detailed discussions of these and other topics see [10], [30], [41], or [57], or almost any book on group theory. It is assumed that the reader is familiar with the elementary group-theo- retical material normally covered in a standard first graduate-level algebra course (in the United States). This includes such concepts as normal sub- group, center, derived (commutator) subgroup, simple group, Sylow sub- group, presentation, etc. The reader is also assumed to be familiar with material from a junior /sen- ior-level linear algebra course, although a fair amount of linear algebra is included in the text as fill-in material, since the content of current linear algebra courses seems not to be very uniform. Any gaps in either group theory or linear algebra can be filled by brows- ing in one or more of the many texts available for such courses. If G is a group and x, y G G we shall write xy to denote the conjugate y~1xy and y x to denote yxy~l. Note that xyz = (xy)z and yz x = y (zx) for all x, y, z £ G. If S is a set we write Perm(S') for the group of all permutations of 5, i.e. all bijections from S to S, with composition of functions as the group operation. If G is a group we say that G acts on S if there is a homomorphism p : G » Perm(5). If tp is one-to-one the action is called faithful Typically G will be a group of linear transformations and S a set of vectors, so if x G G and a G S it will be natural to express the action as a i— (p{x)a, 1
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