Garis-garis Besar
Garis-garis Besar
Perkuliahan
Perkuliahan
15/2/1015/2/10 Sets and RelationsSets and Relations 22/2/10
22/2/10 Definitions and Examples of GroupsDefinitions and Examples of Groups 01/2/10
01/2/10 SubgroupsSubgroups 08/3/10
08/3/10 Lagrange’s TheoremLagrange’s Theorem 15/3/10
15/3/10 Mid-test 1Mid-test 1 22/3/10
22/3/10 Homomorphisms and Normal Subgroups 1Homomorphisms and Normal Subgroups 1 29/3/10
29/3/10 Homomorphisms and Normal Subgroups 2Homomorphisms and Normal Subgroups 2 05/4/10
05/4/10 Factor Groups 1Factor Groups 1 12/4/10
12/4/10 Factor Groups 2Factor Groups 2 19/4/10
19/4/10 Mid-test 2Mid-test 2 26/4/10
26/4/10 Cauchy’s Theorem 1Cauchy’s Theorem 1 03/5/10
03/5/10 Cauchy’s Theorem 2Cauchy’s Theorem 2 10/5/10
10/5/10 The Symmetric Group 1The Symmetric Group 1 17/5/10
17/5/10 The Symmetric Group 2The Symmetric Group 2
22/5/10
Subgroups
Subgroups
Section 2
Definition of a Subgroup
Definition of a Subgroup
A nonempty subset
A nonempty subset
H
H
of a group
of a group
G
G
is called a
is called a
subgroup
subgroup
of
of
G
G
if,
if,
relative to the product
relative to the product
in
in
G
G
,
,
H
H
itself forms a group.
itself forms a group.
A
A
= {1, -1} is a group under the
= {1, -1} is a group under the
multiplication of integers, but is
multiplication of integers, but is
not
not
a
a
subgroup of
subgroup of
viewed as a group with respect
viewed as a group with respect
to +.
Lemma 3
Lemma 3
A nonempty subset
A nonempty subset
H
H
of a group
of a group
G
G
is subgroup if and only if
is subgroup if and only if
H
H
is closed
is closed
with respect to the operation of
with respect to the operation of
G
G
and, given
Examples
Examples
1.1.
The set of all even integers is a subgroup of
The set of all even integers is a subgroup of
the group of integers under +.
the group of integers under +.
Cyclic Subgroup
Cyclic Subgroup
The
The
cyclic subgroup
cyclic subgroup
of
of
G
G
generated by
generated by
a
a
is
is
a set {
a set {
a
a
ii|
|
i any integer}, denoted by (
i any integer}, denoted by (
a
a
).
).
If
If
e
e
is the identity element of
is the identity element of
G
G
, then (
, then (
e
e
)
)
= {
= {
e
e
}.
}.
U
U
nn= (
= (
nn)
)
= (1) = (-1)
= (1) = (-1)
More Examples
More Examples
Let
Let
G
G
be any group. For
be any group. For
a
a
G
G
:
:
The set C(
The set C(
a
a
) = {
) = {
g
g
G
G
|
|
ag
ag
=
=
ga
ga
} is a
} is a
subgroup of
subgroup of
G
G
. It is called the
. It is called the
centralizer
centralizer
of
of
a
a
in
in
G
G
.
.
