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Garis-garis Besar Perkuliahan
15/2/10
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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
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 Z 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 Z is a subgroup of Z 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)

Z = (1) = (-1)

Z7* = (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. If A, B are subgroups of an abelian
group G, show that AB = {ab | a  A, b 
B} is a subgroup of G.
5. Give an example of a group G and two
subgroups A, B of G such that AB is not
a subgroup of G.
6. 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 ^)=