CHAPTER I GROUPS AN D COUNTING PRINCIPLES This chapter sets the stage. We need to define semidirect products and to have rich sets of examples. We discuss groups of permutations and finite groups of rotations. Rather than a dry set of definitions, we take counting principles as our leitmotif and boost the standard Lagrange's theorem (o(H) divides o(G) if H is a subgroup of G) to counting principles for G-spaces. We do this in Section 1.2. Applications are to the classification of finite groups of rotations in Sections 1.4- 5, to the Sylow theorems in Section 1.6, and to group structure in Section 1.7. The important notion of semidirect product is discussed in Section 1.3. 1.1 Groups If you don't already know the definition of a group, you shouldn't be reading this book. Nonetheless, it wouldn't be proper to discuss group representations without defining what a group is so here goes: Definition. A group G is a set with two maps P:GxG-^G the product I : G » G the inverse with shorthand notation P(x,y) = xy, I(x) = x~1. P and I must obey the following axioms: (i) The product is associative: P(x,P(y,z)) = P(P(x,y),z) or, in shorthand, x(yz) = (xy)z for all x,y,z G G. (ii) Existence of an identity: G has a distinguished element e, called the identity, so xe = ex = x for all x 6 G. (iii) Inverses: P(x,I(x)) = P(I(x),x) = e, in shorthand xx"1 = x~xx = e for all x eG. Definition. A group G with commutative product, that is, xy = yx for all x,y G G, is called an abelian group. Definition. If #(G) is finite, we call #(G) the order of G and write it as o(G). l
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