STRUCTURE OF ALGEBRA (Introduction to the Group Theory)
STRUCTURE OF ALGEBRA
(Introduction to the Group Theory)
Drs. Antonius Cahya Prihandoko, M.App.Sc
Mathematics Education Study Program
Definition
A group < G, ∗ > is a set G, together with a binary operation ∗ on G, such that the following axioms are satisfied: 1 G is closed under the operation
∗. That is, ∀a, b ∈ G, a 2 ∗ b ∈ G.
The binary operation ∗ is associative, that is, (∀a, b, c ∈ G), Definition
A group < G, ∗ > is a set G, together with a binary operation ∗ on G, such that the following axioms are satisfied: 1 G is closed under the operation ∗. That is, ∀a, b ∈ G,
a 2 ∗ b ∈ G.
The binary operation ∗ is associative, that is, (∀a, b, c ∈ G), Definition
A group < G, ∗ > is a set G, together with a binary operation ∗ on G, such that the following axioms are satisfied: 1 G is closed under the operation ∗. That is, ∀a, b ∈ G,
a 2 ∗ b ∈ G.
The binary operation ∗ is associative, that is, (∀a, b, c ∈ G), Definition
A group < G, ∗ > is a set G, together with a binary operation ∗ on G, such that the following axioms are satisfied: 1 G is closed under the operation ∗. That is, ∀a, b ∈ G,
a 2 ∗ b ∈ G.
The binary operation ∗ is associative, that is, (∀a, b, c ∈ G), Definition
A group < G, ∗ > is a set G, together with a binary operation ∗ on G, such that the following axioms are satisfied: 1 G is closed under the operation ∗. That is, ∀a, b ∈ G,
a 2 ∗ b ∈ G.
The binary operation ∗ is associative, that is, (∀a, b, c ∈ G), Theorem 1 The identity element in a group is unique.
Theorem 2 The inverse of each element of a group is unique.
Theorem 1 The identity element in a group is unique. Theorem 2 The inverse of each element of a group is unique.
Theorem 1 The identity element in a group is unique. Theorem 2 The inverse of each element of a group is unique.
Theorem 4
If G is a group and a , a , is any n elements of G, then · · · , a
1
2 n
−1 −1 −1 −1(a ∗ a ∗ · · · ∗ a n ) = a ∗ a ∗ · · · ∗ a
1 2 n
1 n−1 . Theorem 4
If G is a group and a , a , is any n elements of G, then · · · , a
1
2 n
−1 −1 −1 −1(a ∗ a ∗ · · · ∗ a n ) = a ∗ a ∗ · · · ∗ a
1 2 n
1 n−1 . Theorem 6
In a group G, the equation ax = b, where a, b ∈ G and x is a −1 variable, has unique solution, that is x
b.
= a
Theorem 7
Theorem 6
In a group G, the equation ax = b, where a, b ∈ G and x is a −1 variable, has unique solution, that is x
b.
= a
Theorem 7
Definition
The result of operation of m factors, a ∗ a ∗ a ∗ a ∗ · · · ∗ a is
m
represented by a ; The result of operation of m factors, −1 −1 −1 −1 −1 −m
a is represented by a ; and
∗ a ∗ a ∗ a ∗ · · · ∗ a
a = e, where e is the identity element in G.
Definition
The result of operation of m factors, a ∗ a ∗ a ∗ a ∗ · · · ∗ a is
m
represented by a ; The result of operation of m factors, −1 −1 −1 −1 −1 −m
a is represented by a ; and
∗ a ∗ a ∗ a ∗ · · · ∗ a
a = e, where e is the identity element in G.
Definition
The result of operation of m factors, a ∗ a ∗ a ∗ a ∗ · · · ∗ a is
m
represented by a ; The result of operation of m factors, −1 −1 −1 −1 −1 −m
a is represented by a ; and
∗ a ∗ a ∗ a ∗ · · · ∗ a
a = e, where e is the identity element in G.
Order of Group The order of a finite group G is the number of elements of G.
If the number of elements of G is infinite, then the order of G is infinite. The order of G is denoted as |G|.
Order of Element
Order of Group The order of a finite group G is the number of elements of G.
If the number of elements of G is infinite, then the order of G is infinite. The order of G is denoted as |G|.
Order of Element
Theorem 1
Let a be an element of a group G. If the order of a is n then there exist n variation of power of a in G, they are
1
2 3 n n−1 a , a , a , , a
· · · , a
Theorem 2 If the order of a is infinite then all power of a are distinct, that is r s
Theorem 1
Let a be an element of a group G. If the order of a is n then there exist n variation of power of a in G, they are
1
2 3 n n−1 a , a , a , , a
· · · , a
Theorem 2
If the order of a is infinite then all power of a are distinct, that is
r s Theorem 1
Let a be an element of a group G. If the order of a is n then there exist n variation of power of a in G, they are
1
2 3 n n−1 a , a , a , , a
· · · , a
Theorem 2
If the order of a is infinite then all power of a are distinct, that is
r s Theorem 1
Let a be an element of a group G. If the order of a is n then there exist n variation of power of a in G, they are
1
2 3 n n−1 a , a , a , , a
· · · , a
Theorem 2
If the order of a is infinite then all power of a are distinct, that is
r s Definition
Let < G, ∗ > be a group and H be a non empty subset of G. H is a subgroup of G if and only if < H,
∗ > is also a group.
Theorem 1 Let < G, ∗ > be a group and H be a non empty subset of G. H
Definition
Let < G, ∗ > be a group and H be a non empty subset of G. H is a subgroup of G if and only if < H,
∗ > is also a group.
Theorem 1
Let < G, ∗ > be a group and H be a non empty subset of G. H
Definition
Let < G, ∗ > be a group and H be a non empty subset of G. H is a subgroup of G if and only if < H,
∗ > is also a group.
Theorem 1
Let < G, ∗ > be a group and H be a non empty subset of G. H
Definition
Let < G, ∗ > be a group and H be a non empty subset of G. H is a subgroup of G if and only if < H,
∗ > is also a group.
Theorem 1
Let < G, ∗ > be a group and H be a non empty subset of G. H
Definition
Let < G, ∗ > be a group and H be a non empty subset of G. H is a subgroup of G if and only if < H,
∗ > is also a group.
Theorem 1
Let < G, ∗ > be a group and H be a non empty subset of G. H
Theorem 2
Let < G, ∗ > be a group and H be non empty subset of G. H is a subgroup of G if it satisfy the two axioms below. 1 Closed. That is 2 −1 (∀c, d ∈ H), c ∗ d ∈ H.
Inverse. That is
(∀c ∈ H), c ∈ H. Theorem 2
Let < G, ∗ > be a group and H be non empty subset of G. H is a subgroup of G if it satisfy the two axioms below. 1 Closed. That is 2 −1 (∀c, d ∈ H), c ∗ d ∈ H.
Inverse. That is
(∀c ∈ H), c ∈ H. Theorem 2
Let < G, ∗ > be a group and H be non empty subset of G. H is a subgroup of G if it satisfy the two axioms below. 1 Closed. That is 2 −1 (∀c, d ∈ H), c ∗ d ∈ H.
Inverse. That is (∀c ∈ H), c ∈ H. Theorem 2
Let < G, ∗ > be a group and H be non empty subset of G. H is a subgroup of G if it satisfy the two axioms below. 1 Closed. That is 2 −1 (∀c, d ∈ H), c ∗ d ∈ H.
Inverse. That is (∀c ∈ H), c ∈ H. Definition
Let < G, ∗ > be a group. Both of H and K are subset of G. Then
H
∗ K = {a ∈ G|a = h ∗ k, h ∈ H ∧ k ∈ K } and −1 −1 Definition
Let < G, ∗ > be a group. Both of H and K are subset of G. Then
H
∗ K = {a ∈ G|a = h ∗ k, h ∈ H ∧ k ∈ K } and −1 −1 Theorem 2
If both of H and K are subgroup of a group < G, ∗ >, then
H ∗ K is also a subgroup if and only if H ∗ K = K ∗ H.
Theorem 3 If both of H and K are subgroup of a group (G, ∗), then H ∩ K is
Theorem 2
If both of H and K are subgroup of a group < G, ∗ >, then
H ∗ K is also a subgroup if and only if H ∗ K = K ∗ H.
Theorem 3
If both of H and K are subgroup of a group (G, ∗), then H ∩ K is
Theorem 2
If both of H and K are subgroup of a group < G, ∗ >, then
H ∗ K is also a subgroup if and only if H ∗ K = K ∗ H.
Theorem 3
If both of H and K are subgroup of a group (G, ∗), then H ∩ K is
Definition
A group G is cyclic if there exists elements a ∈ G such that
m
every element x , where m ∈ G, can be represented by x = a is integer. The element a is called by generator and G is a cyclic group developed by a and denoted :
Definition
A group G is cyclic if there exists elements a ∈ G such that
m
every element x , where m ∈ G, can be represented by x = a is integer. The element a is called by generator and G is a cyclic group developed by a and denoted :
Theorem 2
If G =< a > and b ∈ G then O(b)|O(a).
Theorem 3 Every subgroup of cyclic group
Theorem 2
If G =< a > and b ∈ G then O(b)|O(a).
Theorem 3
Every subgroup of cyclic group
Theorem 2
If G =< a > and b ∈ G then O(b)|O(a).
Theorem 3
Every subgroup of cyclic group
Definition
Let r and s be two positive integers. The positive generator /(d/) of the cyclic group
Theorem 1
If G =< a > is of order non prime n, then every proper
m
subgroup of G is generated by a where m is proper divisor of
n. In converse, if m is a proper divisor of n then G has a proper m subgroup generated by a .
Theorem 1
If G =< a > is of order non prime n, then every proper
m
subgroup of G is generated by a where m is proper divisor of
n. In converse, if m is a proper divisor of n then G has a proper m subgroup generated by a .
Theorem 1
If G =< a > is of order non prime n, then every proper
m
subgroup of G is generated by a where m is proper divisor of
n. In converse, if m is a proper divisor of n then G has a proper m subgroup generated by a .
Order and Power
Let G 1 =< a >, then G may be finite or infinite. 2 If G is infinite, then all power of a are different. Prove it!
If G is finite and of order n, then there are exactly n
different power of a. Why? Order and Power
Let G 1 =< a >, then G may be finite or infinite. 2 If G is infinite, then all power of a are different. Prove it!
If G is finite and of order n, then there are exactly n different power of a. Why? Order and Power
Let G 1 =< a >, then G may be finite or infinite. 2 If G is infinite, then all power of a are different. Prove it! If G is finite and of order n, then there are exactly n different power of a. Why?
Order and Power
Let G 1 =< a >, then G may be finite or infinite. 2 If G is infinite, then all power of a are different. Prove it! If G is finite and of order n, then there are exactly n different power of a. Why?
Theorem 1
The set {0, 1, 2, 3, · · · , n − 1} is the cyclic group Z under
n addition modulo n.
Theorem 2 s
Let G , then b =< a > and |G| = n. If b ∈ G and b = a n
generate a cyclic subgroup H of G containing elements,
Theorem 1
The set {0, 1, 2, 3, · · · , n − 1} is the cyclic group Z under
n addition modulo n.
Theorem 2 s
Let G , then b
=< a > and |G| = n. If b ∈ G and b = a
n
generate a cyclic subgroup H of G containing elements,
Theorem 1
The set {0, 1, 2, 3, · · · , n − 1} is the cyclic group Z under
n addition modulo n.
Theorem 2 s
Let G , then b
=< a > and |G| = n. If b ∈ G and b = a
n
generate a cyclic subgroup H of G containing elements,