文档介绍：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 ex-
presses 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 identity with the properties that, for
any a ∈ G,
ae = ea = a
(3) For any a ∈ G there is a unique element in G denoted a−1 called the inverse of a
such that
aa−1 = a−1a = 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 posed 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 × 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 . T