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Garis-garis Besar Perkuliahan
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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
Homomorphisms and
Normal Subgroups
Homomorphisms
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. 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. 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. 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. 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
-1(w’) is the coset (Ker ) x.
Kernel
Theorem 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. 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. 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) A normal subgroup of a group need not be
characteristic.
4.
If N  G and H is a subgroup of G, show that HN  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) Prove that TbTc = Tbc for all b, c  G [then defines a
homomorphism  of G into A(S)].
b) Describe Ker , the kernel of  : G  A(S).
c) 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?
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please ask questions =(^ y ^)=