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 elem
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 multi
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 mathema
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
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 demonstra
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 i
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 ,
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)
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
resea
Math 3130 Homework 4 Key
Problem 4.50 Take the following permutations in one-line notation and convert them into cycle
notation.
Solution:
(2 1) (1 2)
(1 3 2) (2 3)
(1 3 4 2) (2 3 4)
(2 3 4 5 1) (
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 complet