Download examples of groups

2 | Groups
41
from the context whether we are using addition only or addition and
multiplication. For example, when multiplying matrices with entries
from Zn, we will need both addition modulo n and multiplication
modulo n.
Definition Group
Let G be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (a, b) of elements of G an element in G denoted by ab. We say G is a group under this operation if
the following three properties are satisfied.
1. Associativity. The operation is associative; that is, (ab)c 5 a(bc) for
all a, b, c in G.
2. Identity. There is an element e (called the identity) in G such that
ae 5 ea 5 a for all a in G.
3. Inverses. For each element a in G, there is an element b in G
(called an inverse of a) such that ab 5 ba 5 e.
In words, then, a group is a set together with an associative operation such that there is an identity, every element has an inverse, and any
pair of elements can be combined without going outside the set. Be
sure to verify closure when testing for a group (see Example 5). Notice
that if a is the inverse of b, then b is the inverse of a.
If a group has the property that ab 5 ba for every pair of elements
a and b, we say the group is Abelian. A group is non-Abelian if there
is some pair of elements a and b for which ab 2 ba. When encountering a particular group for the first time, one should determine whether
or not it is Abelian.
Now that we have the formal definition of a group, our first job is
to build a good stock of examples. These examples will be used
throughout the text to illustrate the theorems. (The best way to grasp
the meat of a theorem is to see what it says in specific cases.) As we
progress, the reader is bound to have hunches and conjectures that
can be tested against the stock of examples. To develop a better understanding of the following examples, the reader should supply the
missing details.
EXAMPLE 1 The set of integers Z (so denoted because the German
word for numbers is Zahlen), the set of rational numbers Q (for quotient), and the set of real numbers R are all groups under ordinary addition. In each case, the identity is 0 and the inverse of a is 2a.
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Groups
EXAMPLE 2 The set of integers under ordinary multiplication is not
a group. Since the number 1 is the identity, property 3 fails. For example, there is no integer b such that 5b 5 1.
EXAMPLE 3 The subset {1, 21, i, 2i} of the complex numbers
is a group under complex multiplication. Note that 21 is its own inverse,
whereas the inverse of i is 2i, and vice versa.
EXAMPLE 4 The set Q1 of positive rationals is a group under ordinary multiplication. The inverse of any a is 1/a 5 a21.
EXAMPLE 5 The set S of positive irrational numbers together with 1
under multiplication satisfies the three properties given in the definition
of a group but is not a group. Indeed, "2 ? "2 5 2, so S is not closed
under multiplication.
a b
EXAMPLE 6 A rectangular array of the form c
d is called a
c d
2 3 2 matrix. The set of all 2 3 2 matrices with real entries is a group
under componentwise addition. That is,
c
a 1 a2 b1 1 b2
a b
a1 b1
d
d 1 c 2 2d 5 c 1
c1 1 c2 d1 1 d2
c2 d2
c1 d1
The identity is c
0 0
a b
2a 2b
d , and the inverse of c
d is c
d.
0 0
c d
2c 2d
EXAMPLE 7 The set Zn 5 {0, 1, . . . , n 2 1} for n $ 1 is a group under
addition modulo n. For any j . 0 in Zn, the inverse of j is n 2 j.
This group is usually referred to as the group of integers modulo n.
As we have seen, the real numbers, the 2 3 2 matrices with real entries, and the integers modulo n are all groups under the appropriate addition. But what about multiplication? In each case, the existence of
some elements that do not have inverses prevents the set from being a
group under the usual multiplication. However, we can form a group in
each case by simply throwing out the rascals. Examples 8, 9, and 11
illustrate this.
EXAMPLE 8 The set R* of nonzero real numbers is a group under
ordinary multiplication. The identity is 1. The inverse of a is 1/a.
2 | Groups
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a b
d is the
c d
number ad 2 bc. If A is a 2 3 2 matrix, det A denotes the determinant
of A. The set
EXAMPLE 9† The determinant of the 2 3 2 matrix c
GL(2, R) 5 e c
a b
d ` a, b, c, d
c d
P R, ad
2 bc ? 0 f
of 2 3 2 matrices with real entries and nonzero determinant is a nonAbelian group under the operation
c
a a 1 b1c2 a1b2 1 b1d2
a1 b1 a2 b2
d
d 5 c 1 2
dc
c1a2 1 d1c2 c1b2 1 d1d2 .
c1 d1 c2 d2
The first step in verifying that this set is a group is to show that the
product of two matrices with nonzero determinant also has nonzero
determinant. This follows from the fact that for any pair of 2 3 2
matrices A and B, det (AB) 5 (det A)(det B).
Associativity can be verified by direct (but cumbersome) calcula1 0
a b
tions. The identity is c
d ; the inverse of c
d is
0 1
c d
2b
d
ad 2 bc ad 2 bc
¥
≥
2c
a
ad 2 bc ad 2 bc
(explaining the requirement that ad 2 bc 2 0). This very important
non-Abelian group is called the general linear group of 2 3 2 matrices
over R.
EXAMPLE 10 The set of all 2 3 2 matrices with real number entries
is not a group under the operation defined in Example 9. Inverses do
not exist when the determinant is 0.
Now that we have shown how to make subsets of the real numbers
and subsets of the set of 2 3 2 matrices into multiplicative groups, we
next consider the integers under multiplication modulo n.
†For simplicity, we have restricted our matrix examples to the 2 3 2 case. However,
readers who have had linear algebra can readily generalize to n 3 n matrices.
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Groups
EXAMPLE 11 (L. Euler, 1761) By Exercise 13 in Chapter 0, an
integer a has a multiplicative inverse modulo n if and only if a and n are
relatively prime. So, for each n . 1, we define U(n) to be the set of all
positive integers less than n and relatively prime to n. Then U(n) is a
group under multiplication modulo n. (We leave it to the reader to
check that this set is closed under this operation.)
For n 5 10, we have U(10) 5 {1, 3, 7, 9}. The Cayley table for
U(10) is
mod 10
1
3
7
9
1
3
7
9
1
3
7
9
3
9
1
7
7
1
9
3
9
7
3
1
(Recall that ab mod n is the unique integer r with the property a ? b 5
nq 1 r, where 0 # r , n and a ? b is ordinary multiplication.) In the
case that n is a prime, U(n) 5 {1, 2, . . . , n 2 1}.
In his classic book Lehrbuch der Algebra, published in 1899, Heinrich
Weber gave an extensive treatment of the groups U(n) and described
them as the most important examples of finite Abelian groups.
EXAMPLE 12 The set {0, 1, 2, 3} is not a group under multiplication modulo 4. Although 1 and 3 have inverses, the elements 0 and 2
do not.
EXAMPLE 13 The set of integers under subtraction is not a group,
since the operation is not associative.
With the examples given thus far as a guide, it is wise for the reader
to pause here and think of his or her own examples. Study actively!
Don’t just read along and be spoon-fed by the book.
EXAMPLE 14 For all integers n $ 1, the set of complex nth roots
of unity
e cos
k ? 360°
k ? 360°
1 i sin
` k 5 0, 1, 2, . . . , n 2 1 f
n
n
(i.e., complex zeros of xn 2 1) is a group under multiplication. (See
DeMoivre’s Theorem—Example 7 in Chapter 0.) Compare this group
with the one in Example 3.
2 | Groups
45
The complex number a 1 bi can be represented geometrically as the
point (a, b) in a plane coordinatized by a horizontal real axis and a vertical i or imaginary axis. The distance from the point a 1 bi to the origin is "a 2 1b 2 and is often denoted by Ua 1 bi|. For any angle u, the
line segment joining the complex number cos u 1 i sin u and the origin
forms an angle of u with the positive real axis. Thus, the six complex
zeros of x6 5 1 are located at points around the circle of radius 1, 60°
apart, as shown in Figure 2.1.
Imaginary
– 1 + √3 i
2
2
1 + √3
2 i
2
√3
2
60
–1
1
Real
1
2
– 1 – √3 i
2
2
1 – √3
2
2 i
Figure 2.1
EXAMPLE 15 The set Rn 5 {(a1, a2, . . . , an) U a1, a2, . . . , an [ R}
is a group under componentwise addition [i.e., (a1, a2, . . . , an) 1
(b1, b2, . . . , bn) 5 (a1 1 b1, a2 1 b2, . . . , an 1 bn)].
EXAMPLE 16 For a fixed point (a, b) in R2, define Ta,b: R2 → R2
by (x, y) → (x 1 a, y 1 b). Then G 5 {Ta,b U a, b [ R} is a group
under function composition. Straightforward calculations show that
Ta,bTc,d 5 Ta1c,b1d. From this formula we may observe that G is
closed, T0,0 is the identity, the inverse of Ta,b is T2a,2b, and G is Abelian.
Function composition is always associative. The elements of G are
called translations.
EXAMPLE 17 The set of all 2 3 2 matrices with determinant 1 with entries from Q (rationals), R (reals), C (complex numbers), or Zp (p a prime)
is a non-Abelian group under matrix multiplication. This group is called
the special linear group of 2 3 2 matrices over Q, R, C, or Zp, respectively.
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Groups
If the entries are from F, where F is any of the above, we denote this group
by SL(2, F). For the group SL(2, F), the formula given in Example 9 for
a b
d 2b
d simplifies to c
d . When the matrix
the inverse of c
c d
2c
a
entries are from Zp, we use modulo p arithmetic to compute determinants, matrix products, and inverses. To illustrate the case SL(2, Z5),
3 4
d . Then det A 5 (3 ? 4 2 4 ? 4) mod 5 5
consider the element A 5 c
4 4
4 1
4 24
d. Note
d 5 c
24 mod 5 5 1, and the inverse of A is c
1 3
24
3
1 0
3 4 4 1
d when the arithmetic is done modulo 5.
d 5 c
dc
that c
0 1
4 4 1 3
Example 9 is a special case of the following general construction.
EXAMPLE 18 Let F be any of Q, R, C, or Zp ( p a prime). The set
GL(2, F) of all 2 3 2 matrices with nonzero determinants and entries
from F is a non-Abelian group under matrix multiplication. As in
Example 17, when F is Zp, modulo p arithmetic is used to calculate
determinants, the matrix products, and inverses. The formula given in
a b
d remains valid for elements from
Example 9 for the inverse of c
c d
GL(2, Zp) provided we interpret division by ad 2 bc as multiplication
by the inverse of ad 2 bc modulo p. For example, in GL(2, Z7),
4 5
d . Then the determinant (ad 2 bc) mod 7 is (12 2 30)
consider c
6 3
mod 7 5 218 mod 7 5 3 and the inverse of 3 is 5 [since (3 ? 5)
1 3
3?5 2?5
4 5
d 5 c
d.
d is c
mod 7 5 1]. So, the inverse of c
1?5 4?5
5 6
6 3
1 0
4 5 1 3
d in GL(2, Z7)].
d 5 c
d c
[The reader should check that c
0 1
6 3 5 6
EXAMPLE 19 The set {1, 2, . . . , n 2 1} is a group under multiplication modulo n if and only if n is prime.
EXAMPLE 20 The set of all symmetries of the infinite ornamental
pattern in which arrowheads are spaced uniformly a unit apart along
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2 | Groups
a line is an Abelian group under composition. Let T denote a translation
to the right by one unit, T 21 a translation to the left by one unit, and H a reflection across the horizontal line of the figure. Then, every member of the
group is of the form x1x2 ? ? ? xn, where each xi [
{T, T 21, H}. In this case, we say that T, T 21, and H generate the group.
Table 2.1 summarizes many of the specific groups that we have
presented thus far.
As the examples above demonstrate, the notion of a group is a very
broad one indeed. The goal of the axiomatic approach is to find properties general enough to permit many diverse examples having these
properties and specific enough to allow one to deduce many interesting
consequences.
The goal of abstract algebra is to discover truths about algebraic
systems (that is, sets with one or more binary operations) that are independent of the specific nature of the operations. All one knows
or needs to know is that these operations, whatever they may be, have
Table 2.1 Summary of Group Examples (F can be any of Q, R, C, or Zp; L is a reflection)
Group
Operation
Z
Q1
Addition
Multiplication
0
1
Zn
R*
GL(2, F)
Addition mod n
Multiplication
Matrix
multiplication
0
1
U(n)
Rn
SL(2, F)
Dn
Multiplication
mod n
Componentwise
addition
Matrix
multiplication
Composition
Identity
c
1
0
0
d
1
Form of
Element
k
m/n,
m, n . 0
k
x
a b
c
d
c ,d
ad 2 bc 2 0
k,
gcd(k, n) 5 1
(0, 0, …, 0) (a1, a2, …, an)
1
c
0
d
1
1
0
R0
c
a b
d
c d
ad 2 bc 5 1
Ra, L
Inverse
Abelian
2k
n/m
Yes
Yes
n2k
1/x
Yes
Yes
d
2b
ad 2 bc ad 2 bc
≥
¥
2c
a
ad 2 bc ad 2 bc
Solution to
kx mod n 5 1
(2a1, 2a2, …, 2an)
d
c
2c
2b
d
a
R360 2 a, L
No
Yes
Yes
No
No