Abstract Algebra
Anonymous (For Reddit)Spring 2014
Introduction
The purpose of this text is nothing but personal recreation. At times, I feelthat the best way to improve one’s understanding of a subject is to throwaway the books and exercises, and write down what you remember in amanner that ﬁts your personal intuition. If I have decided to share this withyou, I cannot guarantee that you will beneﬁt from reading this text at anylevel. I do not take responsibility for any errors or inaccuracies in the text.While I do intend to try to write deﬁntions, theorems and proofs in my ownwords, I may have trouble with this, especially if I am unfamiliar with thetopic. Therefore, some parts of the text may be based on
A ﬁrst course in Abstract Algebra
by Fraleigh.
1 Sets, Binary Operations and Groups
A
set
is a collection of objects. Naturally, we are mostly working with numbers, but
{
!
,
@
,
?
}
is still a set, by deﬁnition. We may construct a set asfollows:
{
2
n

n
∈
Z
}
to create the set of all even numbers. An informal deﬁnition of a
binary operation
is an operation on a set, that when performed ontwo members of the set returns another member of the set. As such, addition is not a binary operation on the set
{
1
,
2
,
3
}
, because 2+3 = 5, which isnot in the set. However, addition is a binary operation on the set
{
n

n
∈
Z
}
.A
group
is a set under a binary operation that meets the criteria for abinary operation above, from now on called
closure
. In addition, any groupmust have an identity element
e
, with the property that
a
∗
e
=
e
∗
a
=
a
where
∗
is our binary operation and
a
is any element in our group. Furthermore, for every element
a
∈
G
, where
G
is our group, there must exist aninverse element
a
with the property that
a
∗
a
=
a
∗
a
=
e
. Our last criterion1
is associativity: for any
a,b,c
∈
G
we have that
a
∗
(
b
∗
c
) = (
a
∗
b
)
∗
c
=
a
∗
b
∗
c
.One can play around with examples to get a better grip on the deﬁnition.For example, I claim that
<
Z
,
+
>
is a group, while
<
Z
,
·
>
is not.A
subgroup
H
of a group
G
needs to meet all the axioms above. In addition, all elements in
H
must be in
G
. It follows that
H
and
G
must sharethe identity element and that
H
is closed under the operation and
G
, thatis, for all
a,b
∈
H
we have
a
∗
b
∈
H
and thus
a
∗
b
∈
G
.
1.1 Cyclic Groups
A group
G
is said to be
cyclic
if there is an element
a
∈
G
that
generates
G
, that is,
<
{
a,a
2
,a
3
...
}
,
∗
>
=
G
, where
a
2
=
a
∗
a
. An element with thisproperty is called a
generator
of
G
. While I have, for simplicity, writtenpositive exponents, this also works the other way around (with inverses),and being consciously aware of this may save you from an embarassingmisunderstanding, as showed in this example:
Example 1.1.
Determine if
G
=
<
Z
,
+
>
is cyclic.Note that, given our initial deﬁnition of a cyclic group, where we disregardedinverses,
G
would not be cyclic. However, I claim that 1 is a generator for
G
.Note that
−
1 is the inverse of 1, and we have
{
...
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
...
}
=
Z
as a result of adding both the inverse and the element itself to 1. Thus,
G
is cyclic. A personal anecdote is me answering that
<
Q
,
+
>
is not cyclicbecause it is inﬁnite, which is the right answer with the wrong reason. Thisbrings us to our next example:
Example 1.2.
Determine if
G
=
<
Q
,
+
>
is cyclic.No, and here is the
right
reason: on every interval [
a,b
] there are inﬁnitelymany irrational and rational numbers. Therefore there may not be an element
a
∈
G
that generates the whole group, seeing as some rational numbers(inﬁnitely many, in fact) are skipped in the
a
→
2
a
process.We often see cyclic groups on the form
Z
n
, which can be read as
Z
modulo
n
, where
a
modulo
b
means ”the remainder of
a
when divided by
b
”.Therefore, 1 + 2 = 0 in
Z
3
, as 3 modulo 3 = 0.2
Theorem 1.1.
Every element in a cyclic group
G
generates a subgroup of
G
. Furthermore, this subgroup is also cyclic.
Proof:
3