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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.


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− − ≠ − − .


▪ 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.


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


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=


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.


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..


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.


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.


⊕ 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.


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


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?


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.


MotionMotionMotionMotion SymbolSymbolSymbolSymbol First and Final PositionsFirst and Final PositionsFirst and Final PositionsFirst and Final Positions

No motion e

Rotate 90�

Counterclockwise 90R

Rotate 180�


Rotate 270�


Horizontal flip h

Vertical flip v

Anti-diagonal flip a

Diagonal flip d

Symmetries of a Square

Figure 4


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


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)


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


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?


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?


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


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.


⊕ 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

section 6.2 introduction to groups - mathematics & …farlow/sec62.pdfsection 6.2 introduction...

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