1. Garis-garis Besar Perkuliahan
15/2/10 Sets and Relations
22/2/10 Definitions and Examples of Groups
01/2/10 Subgroups
08/3/10 Lagrange’s Theorem
15/3/10 Mid-test 1
22/3/10 Homomorphisms and Normal Subgroups 1
29/3/10 Homomorphisms and Normal Subgroups 2
05/4/10 Factor Groups 1
12/4/10 Factor Groups 2
19/4/10 Mid-test 2
26/4/10 Cauchy’s Theorem 1
03/5/10 Cauchy’s Theorem 2
10/5/10 The Symmetric Group 1
17/5/10 The Symmetric Group 2
22/5/10 Final-exam

2. Homomorphisms and Normal Subgroups

3. (ab) = (a)(b) for all a, b  G.
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.

4. Examples
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.
Let G be an abelian group and let  : G  G be defined by (x) = x2.
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.

5. 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.

6. 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’.

7. Examples
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.

8. Cayley’s Theorem
Theorem 1. Every group G is isomorphic to some subgroup of A(S), for an appropriate S.
Arthur Cayley ( ) was an English mathematician who worked in matrix theory, invariant theory, and many other parts of algebra.

9. Homomorphism Properties
Lemma 1. If  is a homomorphism of G into G’, then:
(e) = e’, the identity element of G’.
(a-1) = (a)-1 for all a  G.

10. Image and Kernel
Definitions. If  is a homomorphism of G into G’, then:
the image of , (G), is defined by
(G) = {(a) | a  G}.
the kernel of , Ker , is defined by
Ker  = {a | (a) = e’}.

11. Image and Kernel Lemma 2. If  is a homomorphism of G into G’, then:
the image of  is a subgroup of G’.
the kernel of  is a subgroup of G.
if w’  G’ is of the form (x) = w’, then -1(w’) is the coset (Ker ) x.

12. Kernel Theorem 2. If  is a homomorphism of G into G’, then:
Given a  G, a-1(Ker )a  Ker .
 is monomorphism if and only if
Ker  = (e).

13. 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.

14. Examples In Example 8 of Section 1, H = {Ta,b | a rational}  G.
The center Z(G) of any group G is a normal subgroup of G.
In Section 1, the subgroup N = {i, f, f2} is a normal subgroup of S3.

15. Problems
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:
La  A(G)
LaLb = Lab
The mapping  : G  A(G) defined by (a) = La is a homomorphism of G into A(G).

16. Problems M characteristic in G implies that M  G.
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:
M characteristic in G implies that M  G.
M, N characteristic in G implies that MN is characteristic in G.
A normal subgroup of a group need not be characteristic.
If N  G and H is a subgroup of G, show that HN  H.

17. Problems If G is a nonabelian group of order 6, prove that G  S3.
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.
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 .

18. Question? If you are confused like this kitty is,
please ask questions =(^ y ^)=