Section 5: Subgroups
Some notation conventions
+, is usually denoted as +, unless the operation is commutative, in which case is used.
When we write +, we frequently use " to repesent identity, even though that use of " doesn't necessarily
mean the number
section 6
Cyclic Groups
From last section:
Thm Let K be a group and let + K. Then
+ 8 8
is a subgroup of K and is the smallest subgroup that contains +.
Defn: Let K be a group and let + K. Then the subgroup +8 8 of K is called the cyclic
subgroup of K ge
section 4 Groups
Defn: A group
axioms are satsified:
is a set
closed under a binary operation
such that the following
For all
There is an element
For each
in
such that for all
, there is an element
such that
These three requirements are associativity, the
s13
Homomorphisms
Defn
A map
of a group
into a group
is a homomorphism if the homomorphism property
holds for all
Are these homomorphisms?
2.
Let
under addition be given by
, but
, so that
, so, no, this is not a homomorphism.
Let
Since
be given by
the re
s9
Orbits, Cycles, and the Alternating Groups
Each permutation of a set
are in the same cell iff
determines a natural partition of into cells with the property that
for some
. This partition is an equivalence relation, i.e.,
iff
Recall the three condition
s8
Groups of permutations.
Recall a permutation of an ordered set of
for example:
integers is just a rearrangement of that set:
original
rearrangement
1
2
2
3
3
4
4
1
5
5
the text expresses this as
This is a function
such that
and
so we can write:
Since a
s3
Isomorphic Binary Structures
+
,

+ ,
 +
+ ,
, 3.1
,
+
+
,

+ ,
, ,
, ,
, ,
3.5
,
,
,
w
#
$
&
# $
& #
# $
$ &
3.2
&
$
&
#
s
+
,

+ ,
 +
, + ,
3.6
ww
B
C
D
B C
B C
C D
D B
3.3
D
D
B
C
ww
C
B
D
C B
D C
C B
B D
3.4
D
B
D
C
,
+

How are tables 3.1 an
section 14
Factor Groups
Recall this table for
from 10. With our extended knowledge about
cosets, we now know that the table shows the cosets of
of .
is subgroup
mod
is kernel of
Here is a homomorphism. In this section we shall define a coset partition as
Part 1: Groups and Subgroups
section 1: Introduction and Examples
The set of complex numbers is + ,3 + , , with 3# "
Definition:
The absolute value of D + ,3 as kD k k+ ,3k +# ,#
The complex number D + ,3 can be graphed on the Cartesian plane as the orde
Section 2
Binary Operations
Essentially, a binary operation is a rule that "combines" two elements in a set to yield an
element in that same set.
Addition of integers is a binary operation. Thus a$ % b (
while multiplication of integers is another binary
Preliminaries
section 0 Sets & Relations
Assumptions about sets
1.
2.
3.
A set
is made up of elements. If is an element of , then we write
set names are always capital letters. a nonspecific element is represented by a small letter
There is exactly one se
s10 Cosets and the Theorem of Lagrange
Recall the group
is subgroup
Note the partition of
If we denote
into cells
and
, then for each
, we can define a subset
of
as
.
For example,
and
and
Note that
These subsets  indicated by the yellow and gray cells in
section 15
Notation:
Fact:
Factorgroup computations & Simple groups
is a normal subgroup of a group
In the factor group
,
.
is the identity element.
We saw this in the last section, when, for example
is subgroup
mod
is kernel of
the yellow portions of th