1.2 G-SPACES 3 Notice that Tgrg-i = Tg-irg = id so each r is a bijection of S. Thus g »— rg is a group homomorphism of G into the bijections of S and every such homomorphism defines a G-space. r is also called an action of G on S. The language is schizophrenic. The "G" in "G-space" can be specific or can refer to a generic G. So if H is a group, we will often use "G-space for H" rather than //-space. Example 1 (Inner automorphisms). G is a G-space under the action Tx{y) =xyx~l. In fact, rx (y) = xy and rx (y) = yx'1 both define actions which commute with each other and r = r^ o r^2\ It is convenient to extend the notation of group products to subsets of G. Thus, if X C G and y G G, we write yX = {yx | x G G}, and if X, Y C G, then xy = {xy | x e x, y e y}. For a finite group G, let a G G and look at the sequence a 2 ,..., an, For some n m, an = am so a n ~ m _ 1 = a - 1 , that is, a - 1 = afc for A 0. It follows that H C G is a subgroup if if is closed under products that is, H C G is a subgroup if and only if HH = ii . This illustrates the above notation. Note in the above, we have shown that for any a G G, a finite group, there is a smallest n 1 so an e. It is called the order of a. Example 2 (Left cosets). Let H C G be a subgroup. We define G/if, the set of left cosets, to be the distinct elements of {xH \ x G G} that is, G/H is a set of subsets of G. Notice that xH = yH if and only if x~ly G if. The relation x = y ifx~1y G if is easily seen to be an equivalence relation (i.e., x = x for all x, x = y implies y = x and x = y and y = z imply x = 2) precisely because H is a subgroup. The left cosets are precisely the distinct equivalence classes. Define T:GX G/H - G/H by Tg(xH) = gxif. This defines a G-space. One form of the basic counting principle is Lagrange's theorem. Theorem 1.2.1 (Lagrange's theorem). Let H be a subgroup of a finite group G. Then the elements of G/H are disjoint subsets of G whose union is G. Each coset has #(H) elements, so, in particular, #(G) = #(if)#(G/if). In particular, the index ofH, iH = #(G/H), is #(G)/#(H) and #(if) divides #(G). Proof. Since the relation = of Example 2 is an equivalence relation, the cosets are disjoint and their union is all of G. 41)(9) = x 9 is a bijection of G to G, so rx ' H * xH is a bijection and #(H) = #(xH). It follows that #(G) = #(H)#{G/H). D Notice that if a G G and n is the order of a, then H = {e,a,... ,a n _ 1 } is a subgroup, so n = #{H) divides o(G) and, in particular, a°^ = e.
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