2 I. GROUPS AND COUNTING PRINCIPLES We will suppose that you know the basics of point topology as discussed, for example, in Reed-Simon, Vol. 1, Chapter IV [14]. Definition. A topological group G is a Hausdorff topological space which is a group and for which P and / are continuous maps. Chapters I-VI deal almost exclusively with finite groups. Definition. A subgroup H of a group G is a nonempty subset with the property that x,y e H implies that xy and x~l both lie in H. Definition. A homomorphism between groups G and if is a map if : G H so that ip(xy) = ip(x)(p(y). (It follows then that ip(e) = e and tp(x~l) = (p(x)~l.) An isomorphism is a bijection which is a homomorphism. An automorphism is an isomorphism of G to itself. Hom(G, H) will denote the family of homomorphisms from G to H and Aut(G), the automorphisms of G. Note that Aut(G) is a group under the operations that product is composition of maps and inverse is mapping inverse. One large set of automorphisms is defined by the group itself—they are called inner automorphisms. Definition. Let G be a group. For x G G, define ix : G G by ix(y) = xyx"1. ix is called the inner automorphism generated by x. The reader should check that ix is indeed an automorphism of G and note that z i x i ^ %x is a homomorphism of G into Aut(G). Definition. A normal subgroup N of a group G is a subgroup so that ix [N] C N for all x G G. The following is elementary: Proposition 1.1.1. If f : G H is a homomorphism, then Ran/ = {f(x) \ x E G} is a subgroup of H and ker/ = {x G G | /(x) = e#} is a normal subgroup ofG. In the next section, we'll define the important notions of conjugacy class and quotient group. 1.2 G-spaces Many counting arguments involving groups revolve around the fundamental princi- ple for counting when a group acts transitively on a space. So the notion of group action is a critical one. Definition. Let G be a group. A G-space is a set S and a map r : G x S -^ S, so that r(e, s) = s all 5 G S and T ( 0 , T ( M ) ) = T ( 0 M ) (1-2.1) for all g,h G G and s G S. We normally write rg(s) = r(g^s) so r5 : 5 5 and (1.2.1) becomes T ^ = rgh.
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