tailieunhanh - CLASSICAL GEOMETRY

CLASSICAL GEOMETRY — LECTURE NOTES DANNY CALEGARI 1. A CRASH COURSE IN GROUP THEORY A group is an algebraic object which formalizes the mathematical notion which expresses the intuitive idea of symmetry. We start with an abstract definition. Definition . A group is a set G and an operation m : G × G → G called multiplication with the following properties: (1) m is associative. That is, for any a, b, c ∈ G, m(a, m(b, c)) = m(m(a, b), c) and the product can be written unambiguously as abc. (2) There is a unique element e ∈ G called the. | CLASSICAL GEOMETRY LECTURE NOTES DANNY CALEGARI 1. A CRASH COURSE IN GROUP THEORY A group is an algebraic object which formalizes the mathematical notion which expresses the intuitive idea of symmetry. We start with an abstract definition. Definition . A group is a set G and an operation m G X G G called multiplication with the following properties 1 m is associative. That is for any a b c 2 G m a m b c m m a b c and the product can be written unambiguously as abc. 2 There is a unique element e 2 G called the identity with the properties that for any a 2 G ae ea a 3 For any a 2 G there is a unique element in G denoted a 1 called the inverse of a such that . 1 a. 1 a e Given an object with some structural qualities we can study the symmetries of that object namely the set of transformations of the object to itself which preserve the structure in question. Obviously symmetries can be composed associatively since the effect of a symmetry on the object doesn t depend on what sequence of symmetries we applied to the object in the past. Moreover the transformation which does nothing preserves the structure of the object. Finally symmetries are reversible performing the opposite of a symmetry is itself a symmetry. Thus the symmetries of an object also called the automorphisms of an object are an example of a group. The power of the abstract idea of a group is that the symmetries can be studied by themselves without requiring them to be tied to the object they are transforming. So for instance the same group can act by symmetries of many different objects or on the same object in many different ways. Example . The group with only one element e and multiplication e X e e is called the trivial group. Example . The integers Z with m a b a b is a group with identity 0. Example . The positive real numbers R with m a b ab is a group with identity 1. Example . The group with two elements even and odd and multiplication given by the usual rules of addition of .