Chapter 4
Groups
When the group operation is denoted with the symbol
+ then we may denote inverses with negation: a.
In the statement of G2 the language implies that a
group has only one identity element. While this is in
fact true, it remains to be prove
Math 3130, Abstract Algebra
Homework 1 Key
1.1 Suppose that n is a positive whole number greater than 1. Prove that there are an
innite number of positive whole numbers that are not whole-number multiples of n.
Solution: Suppose that nk + 1 is a multiple
Chapter 2
Basic Set Theory
A set is a Many that allows itself to
be thought of as a One.
- Georg Cantor
This chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical
functions. The material is mostly elementary. For
Chapter 3
The Integers and Beyond
God made the integers; all the rest is
the work of man.
- Leopold Kronecker
3.1
Properties of the Integers
We begin with a technical property that is useful in
proofs. While the principle applies to the integers it
is giv
Chapter 1
Logic and Proof
Logic is the beginning of wisdom, not the
end.
-Leonard Nimoy as Commander Spock
n+ 1 fails to appear on the list. This means the list is
not complete. We have thus demonstrated that any
nite list of integers is incomplete and de
Math 3130, Abstract Algebra
Homework 2 Key
3.5Are the two sets the positive multiples of 3 and the positive multiples of 5 the same size?
Solution: Yes. Proof: Recall that two sets are the same size if there is a bijection between them. The
map f (n) = 3n
Math 3130 Homework 4 Key
Problem 4.23 Name the members of the symmetry group of a regular pentagon and give their
Cayley table.
Solution: Names (rotations, ips) cfw_e, r, r2 , r3 , r4 , f1 , f2 , f3 , f4 , f5
e
r
r2
r3
r4
f1
f2
f3
f4
f5
Problem 4.25 Prov
Math 3130, Abstract Algebra
Homework 3 Key
4.1 Prove that in a group that
(ab)1 = b1 a1
Remember: the group may not be commutative.
Solution: Directly check:
(a b) (b1 a1 ) =
=
=
=
(a (b b1 ) a1
(a e) a1
a a1
e
and so we have directly veried the inverse i
Chapter 5
Rings
Nothing proves more clearly that the mind
seeks truth, and nothing reects more glory
upon it, than the delight it takes, sometimes
in spite of itself, in the driest and thorniest
researches of algebra.
- Bernard de Fontenelle
This chapter
Math 3130 Midterm: Due Friday October 26 in Class
Remember to show your work!
Directions This is a take home, open notes test. There are ten problems on the test. You must
answer carefully and completely any ve of the problems. You answer must be your own