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Garis-garis Besar
Perkuliahan
15/2/10
Sets and Relations
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
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
Homomorphisms
and Normal
Subgroups
Homomorphisms
Definition.
Definition Let G, G’ be two groups; then the
mapping : G G’ is a homomorphism if
(ab) = (a)(b) for all a, b G.
The product on the left side—in (ab)—is that of G,
while the product (a)(b) is that of G’.
A homomorphism preserves the operation of G.
Examples
1.
Let G be the group of all positive reals under the
multiplication of reals, and let G’ the group of all
reals under addition. Let : G G’ be defined by
(x) = log10(x) for x G.
2.
Let G be an abelian group and let : G G be
defined by (x) = x2.
3.
Let G be the group of integers under + and G’ = {1,
-1}, the subgroup of the reals under multiplication.
Define (m) = 1 if m is even, (m) = -1 if m is odd.
Homomorphisms
A homomorphism : G G’ is called
monomorphism if
a, b G: a b (a) (b).
A homomorphism : G G’ is called
epimorphism if
a’ G’: a G (a) = a’.
A homomorphism : G G’ is called
isomorphism if it is both 1-1 and onto.
Isomorphic Groups
Two groups G and G’ are said to be
isomorphic if there is an
isomorphism of G onto G’.
We shall denote that G and G’ are
isomorphic by writing G G’.
Examples
4.
Let G be any group and let A(G) be the set of all 1-1
mappings of G onto itself—here we are viewing G
merely as a set, forgetting about its multiplication.
•
Given a G, define Ta : G G by
Ta(x) = ax for every x G.
Verify that Ta Tb = Tab.
•
Define : G A(G) by (a) = Ta for every a G.
Verify that is a monomorphism.
Cayley’s Theorem
Theorem 1.
1 Every group G is isomorphic to
some subgroup of A(S), for an appropriate S.
Arthur Cayley (1821-1895) was an English
mathematician who worked in matrix theory,
invariant theory, and many other parts of
algebra.
Homomorphism Properties
Lemma 1.
1 If is a homomorphism of G
into G’, then:
a)
(e) = e’, the identity element of G’.
b)
(a-1) = (a)-1 for all a G.
Image and Kernel
Definitions.
Definitions If is a homomorphism of G
into G’, then:
a)
the image of , (G), is defined by
(G) = {(a) | a G}.
b)
the kernel of , Ker , is defined by
Ker = {a | (a) = e’}.
Image and Kernel
Lemma 2.
2 If is a homomorphism of G
into G’, then:
a)
the image of is a subgroup of G’.
b)
the kernel of is a subgroup of G.
c)
if w’ G’ is of the form (x) = w’, then (w’) is the coset (Ker ) x.
1
Kernel
Theorem 2.
2 If is a homomorphism of G
into G’, then:
a)
Given a G, a-1(Ker )a Ker .
b)
is monomorphism if and only if
Ker = (e).
Normal Subgroups
Definition.
Definition A subgroup N of G is said to be
a normal subgroup of G if a-1Na N for
every a G.
We write “N is a normal subgroup of G” as
N G.
Theorem 3.
3 N G if and only if every left
coset of N in G is a right coset of N in G.
Examples
1.
In Example 8 of Section 1, H = {Ta,b | a
rational} G.
2.
The center Z(G) of any group G is a
normal subgroup of G.
3.
In Section 1, the subgroup N = {i, f, f2} is
a normal subgroup of S3.
Problems
1.
Let G be any group and A(G) the set of all 1-1
mappings of G, as a set, onto itself. Given a in
G, define La : G G by La(x) = xa-1. Prove that:
a)
La A(G)
b)
LaLb = Lab
c)
The mapping : G A(G) defined by (a) =
La is a homomorphism of G into A(G).
Problems
3.
An automorphism of G is an isomorphism from G to G itself. A
subgroup T of a group G is called characteristic if (T) T for all
automorphisms, , of G. Prove that:
a)
M characteristic in G implies that M G.
b)
M, N characteristic in G implies that MN is characteristic
in G.
c)
4.
A normal subgroup of a group need not be characteristic.
If N G and H is a subgroup of G, show that HN H.
Problems
5.
If G is a nonabelian group of order 6, prove that G S3.
6.
Let G be a group and H a subgroup of G. Let S be the
set of all right cosets of H in G. Define, for b G, Tb : S
S by Tb(Ha) = Hab-1.
a)
b)
c)
Prove that TbTc = Tbc for all b, c G [then defines a
homomorphism of G into A(S)].
Describe Ker , the kernel of : G A(S).
Show that Ker is the largest normal subgroup of G
lying in H [largest in the sense that if N G and N
H, then N Ker .
Question?
If you are confused like this kitty is,
please ask questions =(^ y ^)=
Perkuliahan
15/2/10
Sets and Relations
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
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
Homomorphisms
and Normal
Subgroups
Homomorphisms
Definition.
Definition Let G, G’ be two groups; then the
mapping : G G’ is a homomorphism if
(ab) = (a)(b) for all a, b G.
The product on the left side—in (ab)—is that of G,
while the product (a)(b) is that of G’.
A homomorphism preserves the operation of G.
Examples
1.
Let G be the group of all positive reals under the
multiplication of reals, and let G’ the group of all
reals under addition. Let : G G’ be defined by
(x) = log10(x) for x G.
2.
Let G be an abelian group and let : G G be
defined by (x) = x2.
3.
Let G be the group of integers under + and G’ = {1,
-1}, the subgroup of the reals under multiplication.
Define (m) = 1 if m is even, (m) = -1 if m is odd.
Homomorphisms
A homomorphism : G G’ is called
monomorphism if
a, b G: a b (a) (b).
A homomorphism : G G’ is called
epimorphism if
a’ G’: a G (a) = a’.
A homomorphism : G G’ is called
isomorphism if it is both 1-1 and onto.
Isomorphic Groups
Two groups G and G’ are said to be
isomorphic if there is an
isomorphism of G onto G’.
We shall denote that G and G’ are
isomorphic by writing G G’.
Examples
4.
Let G be any group and let A(G) be the set of all 1-1
mappings of G onto itself—here we are viewing G
merely as a set, forgetting about its multiplication.
•
Given a G, define Ta : G G by
Ta(x) = ax for every x G.
Verify that Ta Tb = Tab.
•
Define : G A(G) by (a) = Ta for every a G.
Verify that is a monomorphism.
Cayley’s Theorem
Theorem 1.
1 Every group G is isomorphic to
some subgroup of A(S), for an appropriate S.
Arthur Cayley (1821-1895) was an English
mathematician who worked in matrix theory,
invariant theory, and many other parts of
algebra.
Homomorphism Properties
Lemma 1.
1 If is a homomorphism of G
into G’, then:
a)
(e) = e’, the identity element of G’.
b)
(a-1) = (a)-1 for all a G.
Image and Kernel
Definitions.
Definitions If is a homomorphism of G
into G’, then:
a)
the image of , (G), is defined by
(G) = {(a) | a G}.
b)
the kernel of , Ker , is defined by
Ker = {a | (a) = e’}.
Image and Kernel
Lemma 2.
2 If is a homomorphism of G
into G’, then:
a)
the image of is a subgroup of G’.
b)
the kernel of is a subgroup of G.
c)
if w’ G’ is of the form (x) = w’, then (w’) is the coset (Ker ) x.
1
Kernel
Theorem 2.
2 If is a homomorphism of G
into G’, then:
a)
Given a G, a-1(Ker )a Ker .
b)
is monomorphism if and only if
Ker = (e).
Normal Subgroups
Definition.
Definition A subgroup N of G is said to be
a normal subgroup of G if a-1Na N for
every a G.
We write “N is a normal subgroup of G” as
N G.
Theorem 3.
3 N G if and only if every left
coset of N in G is a right coset of N in G.
Examples
1.
In Example 8 of Section 1, H = {Ta,b | a
rational} G.
2.
The center Z(G) of any group G is a
normal subgroup of G.
3.
In Section 1, the subgroup N = {i, f, f2} is
a normal subgroup of S3.
Problems
1.
Let G be any group and A(G) the set of all 1-1
mappings of G, as a set, onto itself. Given a in
G, define La : G G by La(x) = xa-1. Prove that:
a)
La A(G)
b)
LaLb = Lab
c)
The mapping : G A(G) defined by (a) =
La is a homomorphism of G into A(G).
Problems
3.
An automorphism of G is an isomorphism from G to G itself. A
subgroup T of a group G is called characteristic if (T) T for all
automorphisms, , of G. Prove that:
a)
M characteristic in G implies that M G.
b)
M, N characteristic in G implies that MN is characteristic
in G.
c)
4.
A normal subgroup of a group need not be characteristic.
If N G and H is a subgroup of G, show that HN H.
Problems
5.
If G is a nonabelian group of order 6, prove that G S3.
6.
Let G be a group and H a subgroup of G. Let S be the
set of all right cosets of H in G. Define, for b G, Tb : S
S by Tb(Ha) = Hab-1.
a)
b)
c)
Prove that TbTc = Tbc for all b, c G [then defines a
homomorphism of G into A(S)].
Describe Ker , the kernel of : G A(S).
Show that Ker is the largest normal subgroup of G
lying in H [largest in the sense that if N G and N
H, then N Ker .
Question?
If you are confused like this kitty is,
please ask questions =(^ y ^)=