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
Outline
1 Group
Group and Its Properties Subgroups
2 Cyclic Groups
Elementary Properties
Outline
1 Group
Group and Its Properties Subgroups
2 Cyclic Groups
Elementary Properties
Group
Definition
A group<G,∗>is a setG, together with a binary operation∗ onG, such that the following axioms are satisfied:
1 Gis 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),
Group
Definition
A group<G,∗>is a setG, together with a binary operation∗ onG, such that the following axioms are satisfied:
1 Gis 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),
Group
Definition
A group<G,∗>is a setG, together with a binary operation∗ onG, such that the following axioms are satisfied:
1 Gis 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),
(∃a−1∈G),a∗a−1=a−1∗a=e, whereeis the identity
Group
Definition
A group<G,∗>is a setG, together with a binary operation∗ onG, such that the following axioms are satisfied:
1 Gis 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),
(∃a−1∈G),a∗a−1=a−1∗a=e, whereeis the identity
Group
Definition
A group<G,∗>is a setG, together with a binary operation∗ onG, such that the following axioms are satisfied:
1 Gis 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), (∃a−1∈G),a∗a−1=a−1∗a=e, whereeis the identity
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,
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,
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,
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
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
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
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
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, whereeis the identity element inG.
Theorem 1
Ifmis a positif integer thena−m= (a−1)m= (am)−1
Theorem 2
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, whereeis the identity element inG.
Theorem 1
Ifmis a positif integer thena−m= (a−1)m= (am)−1
Theorem 2
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, whereeis the identity element inG.
Theorem 1
Ifmis a positif integer thena−m= (a−1)m= (am)−1
Theorem 2
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
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
Order of Group and Elements
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).
Order of Group and Elements
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).
Order of Group and Elements
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).
Order of Group and Elements
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).
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Properties
Theorem 2
IfG=<a>andb∈GthenO(b)|O(a).
Theorem 3
Every subgroup of cyclic group
Corollary
Properties
Theorem 2
IfG=<a>andb∈GthenO(b)|O(a).
Theorem 3
Every subgroup of cyclic group
Corollary
Properties
Theorem 2
IfG=<a>andb∈GthenO(b)|O(a).
Theorem 3
Every subgroup of cyclic group
Corollary
Positive Generator
Definition
Letr andsbe two positive integers. The positive generator /(d/) of the cyclic group
G={nr+ms|n,m∈ Z}
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
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
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
Basic Concept
Order and Power
LetG=<a>, thenGmay be finite or infinite.
1 IfGis infinite, then all power ofaare different. Prove it!
2 IfGis finite and of ordern, then there are exactlyn
different power ofa. Why?
Definition
Basic Concept
Order and Power
LetG=<a>, thenGmay be finite or infinite.
1 IfGis infinite, then all power ofaare different. Prove it!
2 IfGis finite and of ordern, then there are exactlyn
different power ofa. Why?
Definition
Basic Concept
Order and Power
LetG=<a>, thenGmay be finite or infinite.
1 IfGis infinite, then all power ofaare different. Prove it!
2 IfGis finite and of ordern, then there are exactlyn different power ofa. Why?
Definition
Basic Concept
Order and Power
LetG=<a>, thenGmay be finite or infinite.
1 IfGis infinite, then all power ofaare different. Prove it!
2 IfGis finite and of ordern, then there are exactlyn different power ofa. Why?
Definition
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 nd elements, whered isgcdofnands.
Theorem 3
IfG=<a>and|G|=nthen the other generators forGare of
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 nd elements, whered isgcdofnands.
Theorem 3
IfG=<a>and|G|=nthen the other generators forGare of
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 nd elements, whered isgcdofnands.
Theorem 3
