STRUCTURE OF ALGEBRA (Introduction to the Group Theory)

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STRUCTURE OF ALGEBRA

(Introduction to the Group Theory)

Drs. Antonius Cahya Prihandoko, M.App.Sc

Mathematics Education Study Program

Faculty of Teacher Training and Education the University of Jember

Indonesia

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Outline

1 _Group

Group and Its Properties Subgroups

2 _{Cyclic Groups}

Elementary Properties

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Outline

1 _Group

Group and Its Properties Subgroups

2 _{Cyclic Groups}

Elementary Properties

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Group

Definition

A group<G,∗>is a setG, together with a binary operation∗ onG, such that the following axioms are satisfied:

1 _G_{is closed under the operation}_{∗. That is,}_∀a,_b_∈_G,

a∗b∈G.

2 _{The binary operation}_∗_{is associative, that is,}_(∀a,_b,_c_∈_G),

(a∗b)∗c=a∗(b∗c).

3 There is an identity element,e, inG. That is,(∃e∈G),

(∀a∈G),a∗e=e∗a=a.

4 Each element ofGhas own inverse, that is(∀a∈G),

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Group

Definition

1 _G_{is closed under the operation}_{∗. That is,}_∀a,_b_∈_G, a∗b∈G.

2 _{The binary operation}_∗_{is associative, that is,}_(∀a,_b,_c_∈_G),

(a∗b)∗c=a∗(b∗c).

3 There is an identity element,e, inG. That is,(∃e∈G),

(∀a∈G),a∗e=e∗a=a.

4 Each element ofGhas own inverse, that is(∀a∈G),

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Group

Definition

2 _{The binary operation}_∗_{is associative, that is,}_(∀a,_b,_c_∈_G), (a∗b)∗c=a∗(b∗c).

3 There is an identity element,e, inG. That is,(∃e∈G),

(∀a∈G),a∗e=e∗a=a.

4 Each element ofGhas own inverse, that is(∀a∈G),

(∃a−1∈G),a∗a−1=a−1∗a=e, whereeis the identity

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Group

Definition

3 There is an identity element,e, inG. That is,(∃e∈G), (∀a∈G),a∗e=e∗a=a.

4 Each element ofGhas own inverse, that is(∀a∈G),

(∃a−1∈G),a∗a−1=a−1∗a=e, whereeis the identity

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Group

Definition

3 There is an identity element,e, inG. That is,(∃e∈G), (∀a∈G),a∗e=e∗a=a.

4 Each element ofGhas own inverse, that is(∀a∈G), (∃a−1∈G),a∗a−1=a−1∗a=e, whereeis the identity

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Properties of Group

Theorem 1

The identity element in a group is unique.

Theorem 2

The inverse of each element of a group is unique.

Theorem 3

IfGis a group under binary operation∗, thenGsatisfies the left cancelation law and the right cancelation law. That is,

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Properties of Group

Theorem 1

Theorem 2

Theorem 3

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Properties of Group

Theorem 1

Theorem 2

Theorem 3

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Properties of Group

Theorem 4

IfGis a group anda1,a2,· · · ,anis any nelements ofG, then

(a1∗a2∗ · · · ∗an)−1=a−n1∗a−n−11 ∗ · · · ∗a− 1 1

.

Theorem 5

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Properties of Group

Theorem 4

IfGis a group anda1,a2,· · · ,anis any nelements ofG, then

(a1∗a2∗ · · · ∗an)−1=a−n1∗a−n−11 ∗ · · · ∗a− 1 1

.

Theorem 5

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Properties of Group

Theorem 6

In a groupG, the equationax =b, wherea,b∈Gandx is a variable, has unique solution, that isx =a−1b.

Theorem 7

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Properties of Group

Theorem 6

In a groupG, the equationax =b, wherea,b∈Gandx is a variable, has unique solution, that isx =a−1b.

Theorem 7

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Order of Group and Elements

Definition

The result of operation ofmfactors,a∗a∗a∗a∗ · · · ∗ais represented byam; The result of operation ofmfactors, a−1∗a−1∗a−1∗a−1∗ · · · ∗a−1is represented bya−m; and

a0₌_{e, where}_e_{is the identity element in}_G.

Theorem 1

Ifmis a positif integer thena−m= (a−1)m= (am)−1

Theorem 2

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Order of Group and Elements

Definition

Theorem 1

Theorem 2

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Order of Group and Elements

Definition

Theorem 1

Theorem 2

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Order of Group

Theorder of a finite groupGis the number of elements ofG. If the number of elements ofGis infinite, then the order ofGis infinite. The order ofGis denoted as|G|.

Order of Element

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Order of Group and Elements

Order of Group

Theorder of a finite groupGis the number of elements ofG. If the number of elements ofGis infinite, then the order ofGis infinite. The order ofGis denoted as|G|.

Order of Element

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Theorem 1

Letabe an element of a groupG. If the order ofaisnthen there existnvariation of power ofainG, they are

a1,a2,a3,· · · ,an−1_,an

Theorem 2

If the order ofais infinite then all power ofaare distinct, that is ifr 6=s thanar 6=as.

Theorem 3

LetO(a) =n. (ak ₌_e₎_⇔_n|k _{(n is a factor of k).}

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Order of Group and Elements

Theorem 1

a1,a2,a3,· · · ,an−1_,an

Theorem 2

Theorem 3

LetO(a) =n. (ak ₌_e₎_⇔_n|k _{(n is a factor of k).}

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Theorem 1

a1,a2,a3,· · · ,an−1_,an

Theorem 2

Theorem 3

LetO(a) =n. (ak ₌_e)_⇔_n|k _{(n is a factor of k).}

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Order of Group and Elements

Theorem 1

a1,a2,a3,· · · ,an−1_,an

Theorem 2

Theorem 3

LetO(a) =n. (ak ₌_e)_⇔_n|k _{(n is a factor of k).}

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Subgroup

Definition

Let<G,∗>be a group andH be a non empty subset ofG.H is asubgroupofGif and only if<H,∗>is also a group.

Theorem 1

Let<G,∗>be a group andH be a non empty subset ofG.H

is a subgroup ofGif it satisfy these three axioms.

1 Closed.

2 _{Identity element}

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Subgroup

Definition

Theorem 1

Let<G,∗>be a group andH be a non empty subset ofG.H is a subgroup ofGif it satisfy these three axioms.

1 Closed.

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Subgroup

Definition

Theorem 1

1 Closed.

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Subgroup

Definition

Theorem 1

1 Closed.

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Subgroup

Definition

Theorem 1

1 Closed.

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Theorem of Subgroup

Theorem 2

Let<G,∗>be a group andH be non empty subset ofG.H is a subgroup ofGif it satisfy the two axioms below.

1 _{Closed. That is}_(∀c,_d _∈_H_),_c_∗_d _∈_H.

2 _{Inverse. That is}_(∀c _∈_H),_c−1∈H.

Theorem 3

Let<G,∗>be a group andH be a non empty subset ofG. H

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Theorem of Subgroup

Theorem 2

Theorem 3

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Theorem of Subgroup

Theorem 2

Theorem 3

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Theorem of Subgroup

Theorem 2

Theorem 3

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Special Subgroup

Definition

Let<G,∗>be a group. Both ofH andK are subset ofG. Then

H∗K ={a∈G|a=h∗k,h∈H∧k ∈K}

and

H−1₌_{a_∈_G|a₌_h−1_,_h_∈_H}

Theorem 1

If<H,∗>is a subgroup of a group<G,∗>, thenH∗H=H

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Special Subgroup

Definition

Let<G,∗>be a group. Both ofH andK are subset ofG. Then

H∗K ={a∈G|a=h∗k,h∈H∧k ∈K}

and

H−1₌_{a_∈_G|a₌_h−1_,_h_∈_H}

Theorem 1

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Advance Properties

Theorem 2

If both ofHandK are subgroup of a group<G,∗>, then H∗K is also a subgroup if and only ifH∗K =K ∗H.

Theorem 3

If both ofH andK are subgroup of a group(G,∗), thenH∩K is

also a subgroup on<G,∗>.

Theorem 4

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Advance Properties

Theorem 2

Theorem 3

If both ofH andK are subgroup of a group(G,∗), thenH∩K is also a subgroup on<G,∗>.

Theorem 4

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Advance Properties

Theorem 2

Theorem 3

If both ofH andK are subgroup of a group(G,∗), thenH∩K is also a subgroup on<G,∗>.

Theorem 4

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Basic Concept on Cyclic Group

Definition

A groupGiscyclicif there exists elementsa∈Gsuch that every elementx ∈G, can be represented byx =am, wherem is integer. The elementais called bygeneratorandGis a cyclic group developed byaand denoted :

G=<a>

.

Theorem 1

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Basic Concept on Cyclic Group

Definition

A groupGiscyclicif there exists elementsa∈Gsuch that every elementx ∈G, can be represented byx =am, wherem is integer. The elementais called bygeneratorandGis a cyclic group developed byaand denoted :

G=<a>

.

Theorem 1

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Properties

Theorem 2

IfG=<a>andb∈GthenO(b)|O(a).

Theorem 3

Every subgroup of cyclic group

Corollary

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Properties

Theorem 2

Theorem 3

Corollary

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Properties

Theorem 2

Theorem 3

Corollary

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Positive Generator

Definition

Letr andsbe two positive integers. The positive generator /(d/) of the cyclic group

G={nr+ms|n,m∈ Z}

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Properties

Theorem 1

IfG=<a>is of order non primen, then every proper

subgroup ofGis generated byam wheremis proper divisor of n. In converse, ifmis a proper divisor ofnthenGhas a proper subgroup generated byam.

Theorem 2

IfG=<a>andO(a) =nthen|G|=n.

Theorem 3

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Properties

Theorem 1

Theorem 2

Theorem 3

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Properties

Theorem 1

Theorem 2

Theorem 3

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Basic Concept

Order and Power

LetG=<a>, thenGmay be finite or infinite.

1 _If_G_{is infinite, then all power of}_a_{are different. Prove it!}

2 _If_G_{is finite and of order}_{n, then there are exactly}_n

different power ofa. Why?

Definition

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Basic Concept

Order and Power

2 _If_G_{is finite and of order}_{n, then there are exactly}_n

different power ofa. Why?

Definition

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Basic Concept

Order and Power

2 _If_G_{is finite and of order}_{n, then there are exactly}_n different power ofa. Why?

Definition

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Basic Concept

Order and Power

2 _If_G_{is finite and of order}_{n, then there are exactly}_n different power ofa. Why?

Definition

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Properties

Theorem 1

The set{0,1,2,3,· · · ,n−1}is the cyclic groupZnunder

addition modulon.

Theorem 2

LetG=<a>and|G|=n. Ifb ∈Gandb =as, thenb

generate a cyclic subgroupH ofGcontaining n_d elements, whered isgcdofnands.

Theorem 3

IfG=<a>and|G|=nthen the other generators forGare of

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Properties

Theorem 1

addition modulon.

Theorem 2

LetG=<a>and|G|=n. Ifb ∈Gandb =as, thenb generate a cyclic subgroupH ofGcontaining n_d elements, whered isgcdofnands.

Theorem 3

IfG=<a>and|G|=nthen the other generators forGare of

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Properties

Theorem 1

addition modulon.

Theorem 2

LetG=<a>and|G|=n. Ifb ∈Gandb =as, thenb generate a cyclic subgroupH ofGcontaining n_d elements, whered isgcdofnands.

Theorem 3