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15/2/10
22/2/10
01/2/10
08/3/10
15/3/10
22/3/10
29/3/10
05/4/10
12/4/10
19/4/10
26/4/10
03/5/10
10/5/10
17/5/10
22/5/10
Garis-garis Besar
Perkuliahan
Sets and Relations
Definitions and Examples of Groups
Subgroups
Lagrange’s Theorem
Mid-test 1
Homomorphisms and Normal Subgroups 1
Homomorphisms and Normal Subgroups 2
Factor Groups 1
Factor Groups 2
Mid-test 2
Cauchy’s Theorem 1
Cauchy’s Theorem 2
The Symmetric Group 1
The Symmetric Group 2
Final-exam
Subgroups
Section 2
Definition of a Subgroup
A nonempty subset H of a group G is called a
subgroup of G if, relative to the product
in G, H itself forms a group.
A = {1, -1} is a group under the
multiplication of integers, but is not a
subgroup of viewed as a group with respect
to +.
Lemma 3
A nonempty subset H of a group G
is subgroup if and only if H is closed
with respect to the operation of G
and, given a H, then a-1 H.
Examples
1.
The set of all even integers is a subgroup of
the group of integers under +.
2.
Let m > 1 be any integer. The set Hm of all
multiple of m in is a subgroup of under +.
3.
Let a S and let H(a) = {f A(S) | f(a) =
a}. Then H(a) is a subgroup of A(S).
4.
Let G be any group and let a G. The set A =
{ai | i any integer} is a subgroup of G.
Cyclic Subgroup
The cyclic subgroup of G generated by a is
a set {ai | i any integer}, denoted by (a).
If e is the identity element of G, then (e)
= {e}.
Un = (n)
= (1) = (-1)
7* = (3) = (5)
More Examples
Let G be any group. For a G:
The set C(a) = {g G | ag = ga} is a
subgroup of G. It is called the centralizer of
a in G.
The set Z(G) = {z G | xz = zx for all x
G} is a subgroup of G. It is called the
center of G.
Lemma 4
Suppose that G is a group and H is
a nonempty finite subset of G
closed under the product in G.
Then H is a subgroup of G.
Corollary
If G is a finite group and H is a
nonempty subset of G closed under
multiplication in G, then H is a subgroup
of G.
Problems
1.
Find all subgroups of S3.
2.
If G is cyclic, show that every
subgroup of G is cyclic.
3.
If G has no proper subgroups,
prove that G is cyclic of order p,
where p is a prime number.
Problems
4.
5.
6.
If A, B are subgroups of an abelian
group G, show that AB = {ab | a A, b
B} is a subgroup of G.
Give an example of a group G and two
subgroups A, B of G such that AB is not
a subgroup of G.
Let G be a group, H a subgroup of G.
Let Hx = {hx | h H}. Show that, given
a, b G, then Ha = Hb or Ha Hb = .
Question?
If you are confused like this kitty is,
please ask questions =(^ y ^)=
22/2/10
01/2/10
08/3/10
15/3/10
22/3/10
29/3/10
05/4/10
12/4/10
19/4/10
26/4/10
03/5/10
10/5/10
17/5/10
22/5/10
Garis-garis Besar
Perkuliahan
Sets and Relations
Definitions and Examples of Groups
Subgroups
Lagrange’s Theorem
Mid-test 1
Homomorphisms and Normal Subgroups 1
Homomorphisms and Normal Subgroups 2
Factor Groups 1
Factor Groups 2
Mid-test 2
Cauchy’s Theorem 1
Cauchy’s Theorem 2
The Symmetric Group 1
The Symmetric Group 2
Final-exam
Subgroups
Section 2
Definition of a Subgroup
A nonempty subset H of a group G is called a
subgroup of G if, relative to the product
in G, H itself forms a group.
A = {1, -1} is a group under the
multiplication of integers, but is not a
subgroup of viewed as a group with respect
to +.
Lemma 3
A nonempty subset H of a group G
is subgroup if and only if H is closed
with respect to the operation of G
and, given a H, then a-1 H.
Examples
1.
The set of all even integers is a subgroup of
the group of integers under +.
2.
Let m > 1 be any integer. The set Hm of all
multiple of m in is a subgroup of under +.
3.
Let a S and let H(a) = {f A(S) | f(a) =
a}. Then H(a) is a subgroup of A(S).
4.
Let G be any group and let a G. The set A =
{ai | i any integer} is a subgroup of G.
Cyclic Subgroup
The cyclic subgroup of G generated by a is
a set {ai | i any integer}, denoted by (a).
If e is the identity element of G, then (e)
= {e}.
Un = (n)
= (1) = (-1)
7* = (3) = (5)
More Examples
Let G be any group. For a G:
The set C(a) = {g G | ag = ga} is a
subgroup of G. It is called the centralizer of
a in G.
The set Z(G) = {z G | xz = zx for all x
G} is a subgroup of G. It is called the
center of G.
Lemma 4
Suppose that G is a group and H is
a nonempty finite subset of G
closed under the product in G.
Then H is a subgroup of G.
Corollary
If G is a finite group and H is a
nonempty subset of G closed under
multiplication in G, then H is a subgroup
of G.
Problems
1.
Find all subgroups of S3.
2.
If G is cyclic, show that every
subgroup of G is cyclic.
3.
If G has no proper subgroups,
prove that G is cyclic of order p,
where p is a prime number.
Problems
4.
5.
6.
If A, B are subgroups of an abelian
group G, show that AB = {ab | a A, b
B} is a subgroup of G.
Give an example of a group G and two
subgroups A, B of G such that AB is not
a subgroup of G.
Let G be a group, H a subgroup of G.
Let Hx = {hx | h H}. Show that, given
a, b G, then Ha = Hb or Ha Hb = .
Question?
If you are confused like this kitty is,
please ask questions =(^ y ^)=