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 the operation is called in G):
if g, h 2 S then gh 2 S,
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 also should recall that the inverse is unique, so we de
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 correspondence between [m] [n] and [mn]. We dene a function
L(
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 are
given in the exercises. It is a much subtler relatio
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 labeling the face of a
cube by the numbers 1, . . . 6?
Exa
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 set R with distinct elements
0, 1 2 R, and binary operat
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 element g. That is
if n > 0
if n = 0
if n < 0
Note tha
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 the set
cfw_1, 2, . . . , n is a one to one onto functio
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 equal to all others (it exists since the set of common
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 circle has even more. But what does this mean? One way of ex