Chapter 8
Subgroups and matrices
Here is the basic denition
Denition 8.1. A subset S of a group G is called a subgroup if
1. S contains the identity,
2. S is closed under multiplication (or whatever t
Chapter 2
The integers
Recall that an abelian group is a set A with a special element 0, and operation
+ such that
x+0=x
x+y =y+x
x + (y + z) = (x + y) + z
every element x has an inverse x + y = 0
We
Chapter 3
Some nite abelian groups
We want to outline a second proof of theorem 2.12. For this proof, we assume
that X = [m] and Y = [n] for some m, n N. Then we have to construct a
one to one corresp
Chapter 4
Divisibility and
Congruences
Given two integers a and b, we say a divides b or that b is a multiple of a or a|b
if there exists an integer q with b = aq. Some basic properties divisibility a
Chapter 10
Counting Problems
involving Symmetry
Group theory can be applied to counting problems invloving symmetry. Here
are a few such problems.
Example 10.1. How many dice can be constructed by lab
Chapter 5
Commutative Rings and
Fields
The set of integers Z has two interesting operations: addition and multiplication,
which interact in a nice way.
Denition 5.1. A commutative ring consists of a s
Chapter 7
Cyclic groups (revised)
A group (G, , e) is called cyclic if it is generated by
if every element of G is equal to
8
>gg . . . g (n times)
<
n
g = e
> 1 1
:
g g . . . g 1 (|n| times)
a single
Chapter 9
More about Permutations
and Symmetry Groups
Our ultimate goal in this chapter is study some more complicated symmetry
groups than what we did previously. First recall that a permutation of t
Chapter 6
Linear Diophantine
equations
Given two integers a, b, a common divisor is an integer d such that d|a and d|b.
The greatest common divisor is exactly that, the common divisor greater than
or
Chapter 1
The idea of a group
One of our goals is to make precise the idea of symmetry, which is important in
math and other parts of science. Something like a square has a lot of symmetry,
but circl