Download Groups: Introduction, Main Definitions and Examples

Chapter 2
Groups: Introduction, Main
De…nitions and Examples
We begin with some general statements and practical examples, before looking
at precise de…nitions.
2.1
2.1.1
Introduction to Groups
Introduction
The concept of a group really started in the 1830’s; the term was coined by
Evariste Galois in 1830. Today, groups and other algebraic structures are present
in every branch of mathematics as well as other …elds such as chemistry, physics
to name a few.
A group is an algebraic structure consisting of a set and a binary operation.
To be a group, the binary operation must satisfy certain properties we will study
in the next section. The reader has already encountered groups and used their
properties. For example, the set of real numbers with addition is a group. Many
examples include sets of numbers of other mathematical objects. There are also
more applied examples, we present a few in this section. For example, the
possible manipulations of a Rubik’s cube form a group. The examples we will
discuss are related to the symmetries of an object, that is the transformations
which leave the object unchanged.
2.1.2
An Example: Symmetries of a Square
Suppose that we remove a square region from a plane, move it in some way,
then put it back in the place it originally occupied. We wish to describe all
the ways this can be done. We claim that any motion is equivalent to one of
eight possible ways. The square can be placed in four di¤erent positions on two
37
38CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES
di¤erent sides. In fact, the eight motions which will accomplish this are listed
in the table below:
Motion
R0
R90
R180
R270
H
V
D
D0
Description
Rotation of 0
Rotation of 90
Rotation of 180
Rotation of 270
Flip about a horizontal axis
Flip about a vertical axis
Flip about the main diagonal
Flip about the other diagonal
We denote D4 the set of these motions. In other words D4 = (R0 ; R90 ; R180 ; R270 ; H:V; D; D0 ).
It is called the dihedral group of order 8. What is even more amazing is that no
matter how we combine these motions, we always end up with one of them. The
table below, known as a Cayley table after the British mathematician Arthur
Cayley, summarizes these results. It works as follows. We combine elements on
the left …rst column with elements of the top row. The result is placed where the
corresponding row and column intersect. For the time being, let’s denote AB
the combination of A and B with the understanding that B is done …rst. This
is similar to composition of functions. Indeed, these motions can be thought of
as functions from the square region to itself. For example, looking at the table,
we see that HR270 = D0 but R270 H = D.
R0
R90
R180
R270
H
V
D
D0
R0
R0
R90
R180
R270
H
V
D
D0
R90
R90
R180
R270
R0
D
D0
V
H
R180
R180
R270
R0
R90
V
H
D0
D
R270
R270
R0
R90
R180
D0
D
H
V
H
H
D0
V
D
R0
R180
R270
R90
V
V
D
H
D0
R180
R0
R90
R270
D
D
H
D0
V
R90
R270
R0
R180
D0
D0
V
D
H
R270
R90
R180
R0
We make several remarks about the table:
1. By combining these motions, we only obtain one of the eight existing
motions. In other words, we never obtain a new motion. This is one of
the conditions for an operation on a set to form a group. This property is
called closure.
2. We note that if A 2 D4 , then AR0 = R0 A. In other words, R0 does not
change anything. It behaves like 0 for addition of integers or real numbers.
Such an element is called an identity element.
3. We see that for each element A of D4 , there is a unique element B satisfying AB = BA = R0 (the identity element). When this happens, B is
2.1. INTRODUCTION TO GROUPS
39
called the inverse of A. For example, the inverse of H is H. The inverse
of R90 is R270 .
4. We note that the order in which we combine these motions is important.
For example, V R90 = D0 but R90 V = D. So, in general, AB 6= BA if A
and B are elements of D4 . When AB = BA for every A and B, we say
the group is commutative or Abelian (named after Niels Abel).
5. We see that each element of D4 appears only once in each row or column.
We will see it is indeed the case for a group.
6. We see that the table seems pretty orderly. We will see it is indeed the
case for a group.
The properties outlined above will be de…ned precisely in the next section.
They are what makes a group. We will also see that many other properties can
be proven from these.
2.1.3
Problems
Do # 1, 2, 3 on page 35 of your book.
40CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES
2.2
Main De…nitions, Results and Examples
A group, like other mathematical structures you have and/or will study, consists
of a set and one or more operations. For a group, there is one operation, called
a binary operation because it acts on two elements. If G is a set and an
operation, we denote (G; ) the algebraic structure consisting of the set G and
the operation . We say that G is a group under or (G; ) is a group. We
begin by giving some important de…nitions.
2.2.1
De…nitions
De…nition 122 (Binary Operation) A binary operation on a set G is a
function which assigns to each ordered pair of elements of G a third element. G
is said to be closed under the operation if the third element is also an element
of G.
Example 123 R is closed under addition.
Example 124 Z is closed under addition.
Example 125 Z is not closed under division.
Example 126 R+ = fx 2 R j x > 0g is closed under multiplication.
Example 127 R = fx 2 R j x < 0g is not closed under multiplication.
There are several basic properties a binary operations can have. These properties determine what kind of algebraic structure a set together with the binary
operation will be. We list them below.
De…nition 128 (Properties of Binary Operations) Let G be a set and
be a binary operation on G. Let a; b; c be arbitrary elements of G.
1. Commutativity: a b = b a
2. Associativity: a (b c) = (a b) c
3. Right identity: e is a right identity if a e = a
4. Left identity: e is a left identity if e a = a
5. Identity: e is an identity element if a e = e a = a that is a right and
a left identity.
6. Right inverse: a has a right inverse we will call a0 if a a0 = e where e
is an identity element.
7. Left inverse: a has a left inverse we will call a0 if a0
an identity element.
a = e where e is
2.2. MAIN DEFINITIONS, RESULTS AND EXAMPLES
41
8. Inverse: a has an inverse we will call a0 if a0 is a right and a left inverse
of a that is if a a0 = a0 a = e where e is an identity element.
Remark 129 The notion of identity and inverse are dependent on the operation. If there is a risk of confusion, we will specify the operation. For example,
we may talk about the additive identity or the multiplicative identity.
Example 130 The identity element for addition in Z or R is 0.
Example 131 If x is any real number, then
dition.
De…nition 132 (Group) Let G be a set and
x is the inverse of x under ad-
an operation on G.
1. (G; ) is a group if the following properties are satis…ed:
(a) G is closed under
(b)
is associative
(c)
has an identity element
(d) Every element of G has an inverse under
2. If in addition is commutative, (G; ) is said to be a commutative group
or an Abelian group.
It is important to note that it only requires four properties for an algebraic
structure to be a group. As we will see, many more properties can be derived
from these four properties. Hence, every group will satisfy these properties.
2.2.2
Elementary Examples
We now look at some examples. It is important to have many examples in mind
to illustrate the various properties we will study and also to test conjectures we
may have.
Example 133 (Z; +), (Q; +) and (R; +) are groups. In each case the identity
element is 0. The inverse of a is a.
Remark 134 In the above example, we used the terms "the inverse" and "the
identity", thus implying these seem to be unique. Indeed they are, we will prove
this result below.
Example 135 (Z; ) is not a group. Associativity fails.
Example 136 Z is not a group under multiplication. The identity would be
1. The inverse property fails. For example, 3 does not have an inverse under
multiplication in Z.
42CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES
Example 137 R = fx 2 R j x 6= 0g is a group under ordinary multiplication.
1
The identity is 1 and the inverse of a is = a 1 .
a
Example 138 R+ is also a group under ordinary multiplication. The identity
1
is 1 and the inverse of a is = a 1 .
a
Example 139 Q and Q+ are also groups under ordinary multiplication. The
1
identity is 1 and the inverse of a is = a 1 .
a
2.2.3
Results
The properties of a group require the existence of an identity element and also
an inverse for each element in the group. Could there be more than one identity?
Could an element have more than one inverse? We answer these questions here,
and more.
Also, let us make a remark about notation. The properties we are about to
prove are true for every group, no matter which operation the group has. We
will adopt the notation of the multiplication operation. It is important to keep
in mind that what we write applies to whatever operation the group has. When
we write ab we do not mean a "times" b. Rather, we mean a combined with
b using the operation of the group. If the operation is addition, then ab would
mean a + b.
Theorem 140 (Uniqueness of Identity) In a group G, the identity element
is unique.
Proof. One way to prove uniqueness is to assume there are two such elements
and show they are the same. In light of this, let us assume there are two identity
elements e and e0 . We want to show that e = e0 . We know that ae = ea = a
and ae0 = e0 a = a for every a 2 G. So, this works in particular for e and e0 .
We can write
e0
= e0 e since e is an identity element
= e since e0 is an identity element
Therefore, we can now speak of "the" identity element.
Theorem 141 (Cancellation) In a group, the right and left cancellation laws
hold, that is:
ac = bc =) a = b (right cancellation)
ca = cb =) a = b (left cancellation)
2.2. MAIN DEFINITIONS, RESULTS AND EXAMPLES
43
Proof. We only prove the right cancellation law. Suppose that ac = bc and let
c0 be the inverse of c. Then, we have
ac
=
bc =) (ac) c0 = (bc) c0
=)
a (cc0 ) = b (cc0 ) by associativity
=)
ae = be
=)
a=b
One consequence of the cancellation law is that in the Cayley table of a
group, each group element occurs exactly once in each row and each column.
Theorem 142 (Uniqueness of Inverses) Let G be a group. For each a 2 G,
there exists a unique inverse that is a unique element b such that ab = ba = e.
Proof. Suppose a has two inverses, say b and c. We show we must have b = c.
If b and c are inverses of a, then ab = e and ac = e. Therefore ab = ac. By the
cancellation law, we must have b = c.
In light of these properties, we change our notation as follows:
First, recall we will use the multiplicative notation, no matter what the
operation is. So, the inverse of a will be denoted a 1 . Again, keep in mind
that if the operation is addition, then a 1 is really a.
Also, aa:::a will be denoted an (if we have n factors). Again, keep in mind
if the operation is addition, aa:::a really means a + a + ::: + a = na.
The identity element will usually be denoted e. So, a0 = e.
In an abstract group, we do not allow exponents which are not integers.
jnj
If n is negative, we de…ne an = a 1
The usual laws of exponents apply. am an = am+n
n
Be careful, if the group is not commutative, (ab) 6= an bn
As already mentioned, when dealing with a speci…c operation, we must
adapt the notation to it. The table below gives the parallel between multiplication and addition.
Multiplicative group
a b or ab
multiplication
e or 1
identity
a 1
inverse
an
ab 1
Additive group
a+b
addition
0
identity
a
inverse
na
a b
44CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES
Theorem 143 Let G be a group and a and b elements in G. Then,
(ab)
1
=b
1
1
a
1
Proof. Since inverses are unique, and (ab) denotes the inverse of ab, it is
enough to verify that b 1 a 1 is the inverse of ab.
1
(ab) b
a
1
1
= a bb
a
1
1
= aea
1
= aa
= e
and
b
1
a
1
(ab)
= b
1
= b
1
eb
= b
1
b
1
a
a b
= e
These properties are used when solving equations in a group. Probably
until now, all the equations you have solved were with real numbers. You used
techniques without even thinking why and how they worked. It is time to change
this. Whatever you do when solving an equation, you must understand why you
are doing it, if it is even valid. Let us look at some examples.
Example 144 Let a, b, c and x be elements of a group G. Solve the equation
abxc = b for x.
abxc
=
()
()
b () abx = bc
1
bx = a
x=b
1
bc
1
1
bc
a
1
1
There are other important results, which will be proven in the exercises.
We now look at more challenging examples.
2.2.4
More Examples
Example 145 A rectangular array of the form
a
c
b
d
is called a 2
2 ma-
a b
a0 b0
a + a0 b + b0
+
=
. The
0
0
c d
c d
c + c0 d + d0
set of 2 2 matrices with real entries is a group under addition. The identity
0 0
a b
a
b
element is
and the inverse of
is
.
0 0
c d
c
d
trix. Addition is de…ned by
2.2. MAIN DEFINITIONS, RESULTS AND EXAMPLES
Example 146 Multiplication of matrices is de…ned by
a
c
45
b
d
a0
c0
b0
d0
=
aa0 + bc0 bd0 + ab0
. he set of 2 2 matrices with real entries is not a
ca0 + dc0 dd0 + cb0
group under multiplication. To make it a group, we need to add a restriction.
a b
We de…ne the determinant of the 2 2 matrix A =
, denoted det (A),
c d
to be det (A) = ad bc. One of the properties of the determinant is that if A
and B are two 2 2 matrices then det (AB) = det (A) det (B). Let us de…ne the
set GL (2; R), called the general linear group of 2 2 matrices over R by
GL (2; R)
a
c
b
d
j a; b; c; d 2 R and ad
bc 6= 0
Then, GL (2; R) is a non-Abelian group under matrix multiplication.
The iden3
2
b
d
1 0
a b
7
6
tity element is
and the inverse of
is 4 ad c bc ad a bc 5
0 1
c d
ad bc ad bc
Example 147 Let Zn = f0; 1; 2; :::; n 1g for n
1. Zn is a group under
addition modulo n. Recall that if a 2 Zn and b 2 Zn then (a + b) mod n
is the remainder of the division by n in other words if a + b = nq + r then
(a + b) mod n = r. Clearly, we have closure since if a 2 Zn and b 2 Zn then
(a + b) mod n is the remainder of the division by n and is therefore an integer
in Zn . The identity element is 0 and the inverse of j for any j > 0 is n j.
This group is called the group of integers modulo n.
Example 148 Let U (n) be the set of all positive integers less than n which
are relatively prime to n. Then U (n) is a group under multiplication modulo
n. Recall that (ab) mod n is the remainder of the division by n in other words
if ab = nq + r then (ab) mod n = r. Note that if n is prime, then U (n) =
f1; 2; 3; :::; n 1g. For n = 10, U (10) = f1; 3; 7; 9g. We give Cayley’s table for
illustration.
mod 10 1 3 7 9
1
1 3 7 9
3
3 9 1 7
7
7 1 9 3
9
9 7 3 1
The proof that it is a group is left as an exercise.
Example 149 The set Rn = f(a1 ; a2 ; :::; an ) j ai 2 R for i = 1; 2; :::; ng is a
group under componentwise addition that is (a1 ; a2 ; :::; an ) + (b1 ; b2 ; :::; bn ) =
(a1 + b1 ; a2 + b2 ; :::; an + bn ).
Example 150 If F is any of Q, R, C or Zp (p prime), we de…ne the special
linear group of 2 2 matrices over F , denoted SL (2; F ), by
SL (2; F )
a
c
b
d
j a; b; c; d 2 R and ad
bc = 1
46CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES
It is a non-Abelian group under matrix multiplication. Note that in the case of
Zp , we use modulo p arithmetic.
2.3
Problems
1. Prove that if a and n are positive integers then the equation ax mod n = 1
has a solution if and only if 1 = gcd (a; n).
2. Using the above result, prove that U (n) is a group under multiplication
modulo n.
3. Write the Cayley table for the operation of each group below:
(a) Z5 with addition mod 5.
(b) U (5) with multiplication mod 5.
(c) U (10) with multiplication mod 10
a
4. Let G = fa; b; cg and be an operation on G de…ned by the table
b
c
Explain why (G; ) is not a group.
a
a
b
c
b
b
a
b
5. Let a, b, and c be elements of a group. Solve the equation abxc = b for x.
6. Do # 1, 2, 3, 4, 7, 9, 11, 12, 13, 15, 17, 18, 19, 23, 25, 26, 27, 35, 36 on
pages 52-54.
c
c
.
c
a