Section 6.2 Introduction to Groups 1
Section 6.2 Section 6.2 Section 6.2 Section 6.2 Introduction to the Algebraic Introduction to the Algebraic Introduction to the Algebraic Introduction to the Algebraic GroupGroupGroupGroup
Purpose of SectionPurpose of SectionPurpose of SectionPurpose of Section: To introduce the concept of a mathematical structure
called an algebraic group. To illustrate group concepts, we introduce cyclic
and dihedral groups.
IntroductionIntroductionIntroductionIntroduction
The theory of groups is an area of mathematics which is concerned with
underlying relationships of things, and arguably the most powerful tool ever
created for illuminating structure, both mathematical and physical. The word
group was first used by the French genius Evariste Galois in 1830, who wrote
his seminal paper on the unsolvability of the 5th order polynomial equation, the
night before he was killed in a stupid duel at the age of 20. Other early
contributors to the development of group theory were Joseph Louis Lagrange,
(1736-1813), Niels Abel (1802-1829), Augustin-Louis Cauchy (1789-1857),
Arthur Cayley (1821-1895, Camille Jordan (1838-1923), Ludwig Sylow (1832-
1918) and Marius Sophus Lie (1842-1899). Now, merely more than a century
later, group theory has resulted in an amazing unification of areas of
mathematics, including algebra and geometry, long thought to be separate and
unrelated. It is often said that whenever groups make an appearance in a
subject, simplicity is created from chaos. Group theory has played (and is
playing) a crucial role for both chemists and physicists to penetrate the deep
underlying relationships in our amazing world.
Binary Operation and Binary Operation and Binary Operation and Binary Operation and GroupGroupGroupGroupssss
A binary operationbinary operationbinary operationbinary operation on a set A is a rule, which assigns to each pair of
elements of A a unique element of A . Thus, a binary operation is simply a
function :f A A A× → . Two common binary operations familiar to the reader
are ,+ i which assign the sum a b+ ∈� and product a b∈i � to a pair
( ),a b ∈ ×� � of real numbers. We now give a formal definition of a group.
Section 6.2 Introduction to Groups 2
DefinitionDefinitionDefinitionDefinition: An algebraic galgebraic galgebraic galgebraic grouprouprouproup G (or simply groupgroupgroupgroup) is a set of elements with a
binary operationbinary operationbinary operationbinary operation, say “∗ ”, satisfying the closclosclosclosure propertyure propertyure propertyure property
, *a b G a b G∈ ⇒ ∈
as well as the following properties:
▪ Associative:Associative:Associative:Associative: ∗ is associativeassociativeassociativeassociative, that is, for every , ,a b c G∈ , we
have
( ) ( )a b c a b c∗ ∗ = ∗ ∗ .
▪ Identity:Identity:Identity:Identity: G has a unique identityidentityidentityidentity1111 e . That is, for any element a G∈ we
have * *a e e a a= =
▪ Inverse:Inverse:Inverse:Inverse: Every element a G∈ has a unique inverseinverseinverseinverse2222 . That is, for a G∈
there exists an element 1a G
− ∈ that satisfies 1 1* *a a a a e− −= = .
We often denote a group G with operation " "∗ by { },G ∗ .
Often it happens that a b b a∗ = ∗ for all ,a b G∈ . When this happens the group
is called a commutativecommutativecommutativecommutative (or AbelianAbelianAbelianAbelian) group. We often denote the group
operation a b∗ as ab , or maybe something more suggestive like ⊕ if the
group operation is addition or closely resembles addition. A group is called
finitefinitefinitefinite if it contains a finite number of elements and the number of elements in
the group is called the order order order order of the group. If the order of a group G is n , we
denote this by writing G n= If the group is not of finite order we say it is
of infinite orderinfinite orderinfinite orderinfinite order.
In Plain English In Plain English In Plain English In Plain English
▪ Associative:Associative:Associative:Associative: The associative property tells us when we combine three
elements , ,a b c G∈ (keeping them in the same order), the result is unchanged
regardless of which two elements are combined first. There are examples of
algebraic groups where the operation is not associative, the cross product of
vectors in vector analysis is an example of a non-associative operation, as
well as the difference between two numbers3, but by far the majority of binary
operations in mathematics are associative.
1 It is not necessary to state that the identity is unique since it can be proven there is only one identity. In a
more lengthy treatment of groups, we would define the existence of an identity and then prove it is unique,
but here we assume uniqueness to shorten the discussion. 2 Again, it can be proven that the inverse is unique so it is not really necessary to assume uniqueness of an
inverse in the definition. 3 ( ) ( )a b c a b c− − ≠ − − .
Section 6.2 Introduction to Groups 3
▪ Identity:Identity:Identity:Identity: The identity element of a group depends on the binary relation ∗
and is the unique element e G∈ that leaves every element a G∈ unchanged
when combined withe . In the group of the integers � with the binary
operation + (addition), the identity is 0 since 0 0a a a+ = + = for every
integer a . If the binary operation is × (multiplication), then the identity e is 1
since 1 1a a a× = × = for all a in the group.
▪ Inverse:Inverse:Inverse:Inverse: The inverse 1a
− of an element a depends (of course) on the
element a , but also on the identity e and is an element such that when
combined with a yields the identity; that is 1 1aa a a e
− −= = . For example, the
inverse of an integer a with group operation addition + is its negative a− since
( ) ( ) 0a a a a+ − = − + = .
Table 1: Properties of Binary OperationsTable 1: Properties of Binary OperationsTable 1: Properties of Binary OperationsTable 1: Properties of Binary Operations
OperationOperationOperationOperation Associative Associative Associative Associative CommutativeCommutativeCommutativeCommutative IdentityIdentityIdentityIdentity InverseInverseInverseInverse
( ) on P A∪ Yes Yes Yes No
( ) on P A∩ Yes Yes Yes Yes
gcd on � Yes Yes No No
on + � Yes Yes Yes Yes
on − � No No No No
on × � Yes Yes Yes Yes
min on � Yes Yes No No
Example 1: Group Example 1: Group Example 1: Group Example 1: Group Test Test Test Test
Which of the following define a group on the set of integers?
a) { },+� : Integers with the operation of addition.
b) { }, ( ) / 2m n+� : Integers with operation of averaging two integers.
c) { }, −� Integers with operation of taking the difference of two integers.
SolutionSolutionSolutionSolution
a) { },+� : We leave it to the reader to show { },+� is a group. See Problem 5.
b) { }, ( ) / 2m n+� : Taking the average of two integers, say 2 and 3, is not an
integer, hence the averaging operation is not closed in � . Hence � with the
averaging operation is not a group. There is no need to check the other
properties required of a group.
Section 6.2 Introduction to Groups 4
c) { }, −� The integers � with the difference operation is not a group since
subtraction is not associative, i.e. ( ) ( )m n p m n p− − ≠ − − . However, it does
have an identity, 0 since 0 0m m m− = − − = − . Also, every integer has an
inverse, itself, i.e. ( )0, 0m m m m− = − − − = . Nevertheless, failure of the
associative property says it is not a group.
AbstractionAbstractionAbstractionAbstraction Abstraction reveals connections between different areas of mathematics
since the process of abstraction allows one to see essential ideas and see the “forest and
not just the trees.” This broad viewpoint can result in making new discoveries in one
area of mathematics as a result of knowledge in other areas. A disadvantage might be
that highly abstract mathematics is more difficult to master and tends to isolate
mathematics from the outside world.
Cayley Table
The binary operation of a group can be illustrated by means of a Cayley4 table as
drawn in Table 2, which shows the products i j
g g of elements i
g and j
g of a group. It is
much like the addition or multiplication tables the reader studied as a child, except a
Cayley table can record any binary operation. A Cayley table is an example of a latin
square, meaning that every element of the group occurs once and exactly once in every
row and column. We examine the Cayley table to learn about the inner workings of a
group.
∗ 1g e= 2g 3g � jg �
1g e= e 1g 2g � jg �
2g 2g 2
2g 2 3g g � 2 jg g �
� � � � � � �
ig 1i
g g 2ig g 3i
g g � i jg g �
� � � � � � �
Cayley Table for a Group
Figure 2
Example 2 Example 2 Example 2 Example 2
Below we illustrate the only groups of order 2 and 3. Show they are both
commutative. Find the inverse of each element in the group. Show that each
group is associative. Convince yourself that the only groups of these orders
are the ones given. Keep in mind every row and column of the multiplication
4 Arthur Cayley (1821-1895) was an English mathematician
Section 6.2 Introduction to Groups 5
table includes every element of the group exactly once. We leave this fun for
the reader.
Order 2Order 2Order 2Order 2
* e a
e e a
a a e
OOOOrder 3rder 3rder 3rder 3
* e a b
e e a b
a a b e
b b e a
Example Example Example Example 3333:::: The set { }, , ,G a b c d= and binary operation * define a group
illustrated by the Cayley table
a) Is there an identity element? If so, find it.
b) Find the inverse of each element.
c) Is the binary operation commutative?
d) Is ( ) ( )* * * *a b c a b c=
SolutionSolutionSolutionSolution
a) The identity is a since , ,ab ba b ac ca c ad da d= = = = = = .
b) 1 1 1 1, , ,a a b d c c d b− − − −= = = =
c) yes, the multiplication table is symmetric around the main diagonal
d) yes, ( ) ( )* * * and * * *a b c a d d a b c b c d= = = = . In general, there is
no quick way to verify the associative property like there is the commutative
property. You have to check ALL possible arrangements to verify
associativity. On the other hand, if one instance where associativity fails,
then the binary operation * is not associative.
Section 6.2 Introduction to Groups 6
Example Example Example Example 4444: Klein : Klein : Klein : Klein 4444----Group Group Group Group
Show that the set { }, , ,G e a b c= described by the multiplication table in
Figure 1 forms a group. This group is called the KleinKleinKleinKlein5555 4444----groupgroupgroupgroup, which is the
symmetry group of a (non square) rectangle6 studied in Section 6.1. There
are exactly two distinct groups of order four, the Klein 4-group and the cyclic
group 4� which we will study shortly.
∗ e a b c
e e a b c
a a e c b
b b c e a
c c b a e
Multiplication Table for the Klein four-group
Figure 1
Proof: Proof: Proof: Proof: First observe all products of elements of G belong to G since the
table consists only of elements of G . The hardest requirement to check is
associativity, which requires we check ( ) ( )r s t r s t∗ ∗ = ∗ ∗ , where ,r s can be
any of the elements , , ,e a b c , which means we have 34 48= equations to check
since each or the , ,r s t in the associative formula can take on one of four
values , , ,e a b c . The computations can be simplified by observing the group is
commutative (i.e. r s s r∗ = ∗ for all ,r s G∈ ), which implies ( ) ( )r r r r r r∗ ∗ = ∗ ∗
so we have associativity when r s t= = . Other shortcuts tricks can be used
(as well as computer algebra systems) to shorten the list of elements you
must check. In this example, we observe that the group operation ∗ is simply
the composition of functions and we can resort to the fact that composition of
functions is associative.
Finding the group identity e is easy since multiplying any member of the
group by e , either on the left of right, does not change the member. You can
see that the first row and column of the table are the same as the group
elements themselves.
5 Felix Klein (1849-1925) was a German geometer and one of the major mathematicians of the 19
th
century. 6 Try interpreting the elements ,e a of the Klein group are 0 and 180 degree rotations of a rectangle, and
,c d the horizontal and vertical flips of the rectangle..
Section 6.2 Introduction to Groups 7
Finally, to find the inverse 1r− of an element r simply follow along the
row labeled " "r until you get to the group identity e , then the inverse 1r− is
the column label above e . You could also do the same thing by going down
the column labeled " "r until reaching e , then the row label at the left of e is 1r− . In the Klein four-group each element 1, , ,a b c is its own inverse since the
identity e lies along the diagonal of the “multiplication” table.
Familiar Familiar Familiar Familiar GroupsGroupsGroupsGroups
You are familiar with more groups that you probably realize. Table 2
shows just a few algebraic groups you might have seen in earlier studies.
GroupGroupGroupGroup ElementsElementsElementsElements OperationOperationOperationOperation IdentityIdentityIdentityIdentity InverseInverseInverseInverse AbelianAbelianAbelianAbelian
� n ∈� addition 0 n− yes
+� /
, 0
m n
m n > multiplication
1
/n m yes
n� { }0,1, 2,..., 1k n∈ −
multiplication
mod n 0 n k− yes
{ }0−� x nonzero real
number multiplication 1 1/ x yes
2� ( ) 2,a b ∈� vector
addition ( )0,0 ( ),a b− − yes
( )2,
general linear
group
GL �
0
a b
c d
ad bc
− ≠
matrix
multiplication
1 0
0 1
d b
ad bc ad bc
c a
ad bc ad bc
− − −
− − −
no
( )2,
special linear
group
SL �
1
a b
c d
ad bc
− =
matrix
multiplication
1 0
0 1
d b
c a
− −
no
Common Groups
Table 2
Notational Note:Notational Note:Notational Note:Notational Note: Repeated multiplication of an element g of a group by itself
result in powers of an element and are denoted by , 1,2,...ng n = . When 0n =
we define 0g as the identity 0g e= .
Cyclic Groups (Cyclic Groups (Cyclic Groups (Cyclic Groups (Modular Arithmetic)Modular Arithmetic)Modular Arithmetic)Modular Arithmetic)
The most common and most simple of all groups are the cyclic groupscyclic groupscyclic groupscyclic groups,
which are well-known to every child who has learned to keep time.
Section 6.2 Introduction to Groups 8
Definition:Definition:Definition:Definition: A finite cyclic group ( ),nZ ∗ of order n is a group that contains an
element n
g Z∈ called the generatorgeneratorgeneratorgenerator of the group, such that
{ }2 3 1, , , ,..., n
ng e g g g g Z−≡ = .
where “powers” of g are simply repeated multiplications7 of g ; that is 2 3 2, ,...g g g g g g= ∗ = ∗
For example, the three rotational symmetries { }120 240, ,e R R of an equilateral
triangle form a cyclic group 3� with generator 120g R= (note that 2 3
120 240 120,R R R e= = ).
Cyclic groups also describe modular (or clock) arithmetic, which is the
type of arithmetic we carry out when keeping time on 24-hour timepiece,
where the numbers “wrap around” after 24 hours. However, unless you are
in the military, your clock lists hours from 0 to 11 so telling time is done, as
they say, modulo 12, as in 5 hours after 9 P.M. is ( )5 9 mod(12) 3 A.M.+ = .
This leads us to the cyclic group 12Z with elements
{ }12 0,1, 2,3,4,5,6,7,8,9,10,11Z =
and the group operation on 12Z exactly what you do when you keep time, that
is
( )mod12a b a b⊕ = +
where “mod 12” refers to computing a b⊕ by computing ( )a b+ then taking
its remainder after dividing by 12. (We denote the group operation by ⊕ to
remind us it is addition, only reduced modulo 12.) For example
( )2 9 5 mod12= + which, related to a 12-hour clock, translates into 2 A.M is 5
hours after 9 P.M. The hours of the clock 12Z and keeping time using this
binary operation defines an Abelian group of order 12, called a cyclic groupcyclic groupcyclic groupcyclic group,,,,
whose Cayley table is shown in Table 3.
7 We use the word “multiplication” here, but keep in mind the group operation can mean any binary
operation, even addition.
Section 6.2 Introduction to Groups 9
⊕ 0 1 2 3 4 5 6 7 8 9 10 11
0 0 1 2 3 4 5 6 7 8 9 10 11
1 1 2 3 4 5 6 7 8 9 10 11 0
2 2 3 4 5 6 7 8 9 10 11 0 1
3 3 4 5 6 7 8 9 10 11 0 1 2
4 4 5 6 7 8 9 10 11 0 1 2 3
5 5 6 7 8 9 10 11 0 1 2 3 4
6 6 7 8 9 10 11 0 1 2 3 4 5
7 7 8 9 10 11 0 1 2 3 4 5 6
8 8 9 10 11 0 1 2 3 4 5 6 7
9 9 10 11 0 1 2 3 4 5 6 7 8
10 10 11 0 1 2 3 4 5 6 7 8 9
11 11 0 1 2 3 4 5 6 7 8 9 10
Cayley table for the cyclic group of 12 elements
Table 3
Figure 3 shows various clocks that give rise to different cyclic groups.
The cyclic group 6Z consists of elements { }6 1 2 3 4 50, , , , ,Z r r r r r= were as always
e is the identity map and j
r is rotation of the clock by 60 j degrees, 1,...,5j = .
This set, along with the binary operation of doing one operation after another
(function composition) forms a group with the following Cayley table. In other
to make the table read faster, we have replaced the angles of rotation j
r by
the time on the hour hand, 1,2,3,… , 11.
Section 6.2 Introduction to Groups 10
Cyclic Groups n�
2Z
Cyclic
Group
Of
Order
2
⊕ 0 6
0 0 6
6 6 0
3Z
Cyclic
Group
of
Order
3
⊕ 0 4 8
0 0 4 8
4 4 8 0
8 8 0 4
4Z
Cyclic
Group
of
Order
4
⊕ 0 3 6 9
0 0 3 6 9
3 3 6 9 0
6 6 9 0 3
9 9 0 3 6
6Z
Cyclic
Group
of
Order
6
⊕ 0 2 4 6 8 10
0 0 2 4 6 8 10
2 2 4 6 8 10 0
4 4 6 8 10 0 2
6 6 8 10 0 2 4
8 8 10 0 2 4 6
10 10 0 2 4 6 8
12Z
Cyclic
of
Order
12
See
Table 1
Cyclic Groups
Figure 3
Section 6.2 Introduction to Groups 11
Example 5 (Symmetries of a Square) Example 5 (Symmetries of a Square) Example 5 (Symmetries of a Square) Example 5 (Symmetries of a Square) Figure 4 shows the eight symmetries
of a square, called the octic octic octic octic group.
a) Is the octic group commutative? Hint: Compare products 270R a
and 270 aR .
b) There are several subsets of the eight symmetries that form a group
in their own right. These are called subgroups of the octic group. Can you
find all ten of them?
SolutionSolutionSolutionSolution
a) The reader can check but 270 70R a aR≠ . Hence, the octic group is not
commutative.
b) The 10 subgroups of the octic group are
{ } { } { } { } { } { } { } { }180 180 180 4, , , , , , , , , , , , , , , , . , ,e e v e h e d e a e R e R v h e R d a D
The reader can visualize these symmetries and make Cayley tables for them.
(See Problem 16.) These subgroups form a partially ordered set and can be
put in the Hasse diagram shown in Figure 5.
Section 6.2 Introduction to Groups 12
MotionMotionMotionMotion SymbolSymbolSymbolSymbol First and Final PositionsFirst and Final PositionsFirst and Final PositionsFirst and Final Positions
No motion e
Rotate 90�
Counterclockwise 90R
Rotate 180�
Counterclockwise 180R
Rotate 270�
Counterclockwise 270R
Horizontal flip h
Vertical flip v
Anti-diagonal flip a
Diagonal flip d
Symmetries of a Square
Figure 4
Section 6.2 Introduction to Groups 13
Hasse Diagram for the Subgroups of the Octic Group 4D
Figure 5
Symmetry GroupsSymmetry GroupsSymmetry GroupsSymmetry Groups of of of of n ----gons: Dihedral Groupsgons: Dihedral Groupsgons: Dihedral Groupsgons: Dihedral Groups
In Section 6.1 we saw how symmetries of a figure in the plane, which is
a rigid motion which leaves the figure unchanged, can create new symmetries
by following one symmetry after another. In this way, one creates an
“arithmetic” of symmetries, where the composition of symmetries plays the
role of multiplication, and the system contains identity elements, inverses and
all the goodies on an “arithmetic” system. In other words, a group of
symmetries, called the symmetrsymmetrsymmetrsymmetryyyy group group group group of the figure. Every figure no matter
how “non symmetric” has at least one symmetric group, namely the group
consisting only of the identity or ‘do nothing symmetry.” The more
“symmetric” a figure the more elements in its symmetry group. In Section 6.1
we saw that the (non square) rectangle had four symmetries; namely rotations
Section 6.2 Introduction to Groups 14
of 0 and 180 degrees, and a horizontal and vertical flip about the midlines,
which constitute the Klein 4-group. On the other hand, the more “symmetric”
square has 8 symmetries in its symmetry group. (Can you find them?) Some
figures have both rotational and flip symmetries.
A polygon is called regularregularregularregular if all its sides have the same length and all its
angles are equal. An equilateral triangle is a regular 3-gon, a square is a
regular 4-gon, a pentagon is a regular 5-gon and so on. The symmetry group
of a regular n -gon, which has n rotational and n flip symmetries, for a total
of 2n symmetries, is called the dihedral groupdihedral groupdihedral groupdihedral group of the n -gon and denoted by
nD Can you find the 10 symmetries of the dihedral group 5D of the pentagon
drawn in Figure 5?
Find the Symmetric Group 5D
Figure 5
We are getting ahead of ourselves, but in addition to the complete dihedral
group of 10 symmetries of a pentagon, there is also a “smaller” group of five
rotation symmetriesrotation symmetriesrotation symmetriesrotation symmetries , called a subgroupsubgroupsubgroupsubgroup, which is a group in its own right, of
the larger group of 10 symmetries as well as a subgroup of the 5 flip
symmetries. Figure 6 shows commercial figures whose symmetry groups are
the dihedral groups 1 5D D− .
2 symmetries
1D one rotation (0 degrees), one vertical
flip
4 symmetries
2D two rotations, two flips (horizontal
and vertical axes)
Section 6.2 Introduction to Groups 15
6 symmetries
3D three rotations, three flips
8 symmetries
4D four rotations, four flips
10
symmetries
5D five rotations, five flips
Figure with Dihedral Symmetry Groups
Figure 6
Section 6.2 Introduction to Groups 16
ProblemsProblemsProblemsProblems
1. Do the following sets with given binary operations form a group? If it
does, give the identity element and the inverse of each element. If it does not
form a group, say why.
a) All even numbers, addition
b) { }1,1− , multiplication
c) All positive real numbers, multiplication
d) All nonzero real numbers, division
e) All 2 2× real matrices, matrix addition
f) the four numbers 1, 1, ,i i− − where 1i = − is the unit
complex number, the binary operation is ordinary multiplication.
g) { }, ,a b c with operation ∗ defined by the Cayley table
∗ a b c
a b c a
b a b c
c c a b
2. (Finish the Group)(Finish the Group)(Finish the Group)(Finish the Group) Complete the following Cayley table for a group of
order three.
∗ e a b
e e a b
a a
b b
3. (Finish the Group)(Finish the Group)(Finish the Group)(Finish the Group) Complete the following Cayley table for a group of
order four without looking at the Cayley tables of the Klein 4- group or the
cyclic group of order 4 in the text.
∗ e a b c
e e a b c
a a
b b
c c
4. (Verification of a Group) (Verification of a Group) (Verification of a Group) (Verification of a Group) Do the nonzero integers with the operation of
multiplication form a group?
Section 6.2 Introduction to Groups 17
5. (Group You are Well Familiar)(Group You are Well Familiar)(Group You are Well Familiar)(Group You are Well Familiar) Show that { },+� is a group (i.e. the
integers with the operation of addition) is a group.
6. (Property of a Group)(Property of a Group)(Property of a Group)(Property of a Group) Verify that for all elements ,a b in a group the
identity ( )1 1 1
ab b a− − −= holds. Hint: Show that ( ) ( )1 1ab a b e− − = .
General Properties of GroupsGeneral Properties of GroupsGeneral Properties of GroupsGeneral Properties of Groups
7. Show that a group has exactly one identity.
8. Show that every element in a group has no more than one inverse.
9. Show that in a group the identity ( )( )( ) ( )( )ab c d ab cd= holds.
10. Let , ,a b c are members of a set with a binary operation ∗ on the set. If
a b= , then the multiplication rule says ca cb= .
a) Show that in a group we can cancel the c .
b) Given an example of elements in the set { }6 0,1, 2,3,4,5=� with
addition modulo 6, where cancellation does not hold.
11. (Cyclic Group) (Cyclic Group) (Cyclic Group) (Cyclic Group) The cyclic group of order 6 describes the rotational
symmetries of a regular hexagon. Its Cayley table is shown in Table 4.
For notational simplicity the group elements are the number of degrees
required to map the hexagon back onto itself.
⊕ 0� 60� 120� 180� 240� 300�
0� 0� 60� 120� 180� 240� 300�
60� 60� 120� 180� 240� 300� 0�
120� 120� 180� 240� 300� 0� 60�
180� 180� 240� 300� 0� 60� 120�
240� 240� 300� 0� 60� 120� 180�
300� 300� 0� 60� 120� 180� 240�
Cayley Table for 6�
Table 4
a) What is the inverse of each element of the group?
b) The order of an elementorder of an elementorder of an elementorder of an element of a group is defined as the (smallest)
number of repeated operations on itself that results in the group identity.
What is the order of each element of the group?
Section 6.2 Introduction to Groups 18
c) Show that by taking repeated operations with itself of the element
240� the set of elements obtained is itself a group. Construct the Cayley
table of this group.
12. (Affine Group)(Affine Group)(Affine Group)(Affine Group) The set G of all transformations from the plane to the p
lane of the form
x ax by c
y dx ey f
′ = + +
′ = + +
where , , , , , and a b c d e f are real numbers satisfying 0ad bc− ≠ , is a group if we
define the group operation of performing one operation after the other, this
forms a group called the affine groupaffine groupaffine groupaffine group.
a) What is the identity of the affine group? That is, what are the values
of , , , , ,a b c d e f ?
b) What is the inverse of ( )0,0 ?
13. (Mod 5 Multiplication). (Mod 5 Multiplication). (Mod 5 Multiplication). (Mod 5 Multiplication) Create the multiplication table for the integers
0,1, 2,3,4 where multiplication defined as ( )mod 5 arithmetic. Show this defines
a group.
14. (Mod (Mod (Mod (Mod 4444 Multiplication) Multiplication) Multiplication) Multiplication) Create the multiplication table for the integers
0,1,2,3 for modular arithmetic ( )mod 4 and show that this does not define a
group. In other that the numbers 0,1, 2,..., 1n − forms a group under
( )mod n multiplication, it must be true that n is a prime number.
15. ((((Subgroups of the Subgroups of the Subgroups of the Subgroups of the Dihedral Group Dihedral Group Dihedral Group Dihedral Group 4D )))) A subgroup of a group is a subset
of the elements of a group which itself a group using the group operation of the
larger group. The Hasse diagram for the subgroups of the symmetries of a
square (i.e. the dihedral group ) is shown in Figure 6. The letter " "F
represents the identity element, the other letters represent rotations and
reflections of F . Interpret the rotations in the subgroups and make a Cayley
table for them.
16. (Subgroups of the Octic Group)(Subgroups of the Octic Group)(Subgroups of the Octic Group)(Subgroups of the Octic Group) Make a Cayley table for the subgroups of
the octic group.
a) { },e v
b) { },e h
c) { },e d
Section 6.2 Introduction to Groups 19
d) { },e a
e) { }180,e R
f) { }180, , ,e R v h
g) { }180, , ,e R d a
Hasse diagram for the Subgroups of 4D
Figure 6
17. (Isomorphic Groups)(Isomorphic Groups)(Isomorphic Groups)(Isomorphic Groups) Sometime two groups appear different but are really
the same group. The two groups in Figures 7, called Group A and Group B
look different, but are really the same, or what are called isomorphic isomorphic isomorphic isomorphic groups.
Convince yourself the two groups are the same by making the substitution, or
isomorphismisomorphismisomorphismisomorphism
, 0 , 1 , 2 , 3i a b c⊕ → ⊗ → → → →
that sends Group A into Group B.
Section 6.2 Introduction to Groups 20
⊕ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
Group A
⊗ i a b c
i i a b c
a a b c i
b b c i a
c c i a b
Group B
18. (Harder to See Isomorphic Groups)(Harder to See Isomorphic Groups)(Harder to See Isomorphic Groups)(Harder to See Isomorphic Groups) Show that Group C and Group D are
isomorphic (the same group) by making the substitution
, 0 1, 1 2, 2 4, 3 3⊕ → ⊗ → → → →
in Group C, and then interchanging the 3rd and 4th columns, followed by the 3rd
and 4th rows of the resulting table.
⊕ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
Group C
⊗ 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
Group D