This preview shows pages 1–3. Sign up to view the full content.
Chapter 1
Group Fundamentals
1.1 Groups and Subgroups
1.1.1 Defnition
A
group
is a nonempty set
G
on which there is defned a binary operation (
a,b
)
→
ab
satisFying the Following properties.
Closure
:
IF
a
and
b
belong to
G
, then
ab
is also in
G
;
Associativity
:
a
(
bc
)=(
ab
)
c
For all
a,b,c
∈
G
;
Identity
:
There is an element 1
∈
G
such that
a
1=1
a
=
a
For all
a
in
G
;
Inverse
:
IF
a
is in
G
, then there is an element
a
−
1
in
G
such that
aa
−
1
=
a
−
1
a
=1.
A group
G
is
abelian
iF the binary operation is commutative, i.e.,
ab
=
ba
For all
in
G
. In this case the binary operation is oFten written additively ((
)
→
a
+
b
), with
the identity written as 0 rather than 1.
There are some very Familiar examples oF abelian groups under addition, namely the
integers
Z
, the rationals
Q
, the real numbers
R
, the complex numers
C
, and the integers
Z
m
modulo
m
. Nonabelian groups will begin to appear in the next section.
The associative law generalizes to products oF any fnite number oF elements, For exam
ple, (
ab
)(
cde
)=
a
(
bcd
)
e
. A Formal prooF can be given by induction. IF two people A and
B Form
a
1
···
a
n
in di±erent ways, the last multiplication perFormed by
A
might look like
(
a
1
a
i
)(
a
i
+1
a
n
), and the last multiplication by B might be (
a
1
a
j
)(
a
j
+1
a
n
).
But iF (without loss oF generality)
i<j
, then (induction hypothesis)
(
a
1
a
j
a
1
a
i
)(
a
i
+1
a
j
)
and
(
a
i
+1
a
n
a
i
+1
a
j
)(
a
j
+1
a
n
)
.
By the
n
= 3 case, i.e., the associative law as stated in the defnition oF a group, the
products computed by A and B are the same.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
CHAPTER 1. GROUP FUNDAMENTALS
The identity is unique (1
0
=1
0
1 = 1), as is the inverse of any given element (if
b
and
b
0
are inverses of
a
, then
b
b
=(
b
0
a
)
b
=
b
0
(
ab
)=
b
0
1=
b
0
). Exactly the same argument
shows that if
b
is a right inverse, and
b
0
a left inverse, of
a
, then
b
=
b
0
.
1.1.2 Defnitions and Comments
A
subgroup
H
of a group
G
is a nonempty subset of
G
that forms a group under the
binary operation of
G
. Equivalently,
H
is a nonempty subset of
G
such that if
a
and
b
belong to
H
,sodoes
ab
−
1
. (Note that 1 =
aa
−
1
∈
H
; also,
ab
=
a
((
b
−
1
)
−
1
)
∈
H
.)
If
A
is any subset of a group
G
, the
subgroup generated by
A
is the smallest subgroup
containing
A
, often denoted by
h
A
i
. Formally,
h
A
i
is the intersection of all subgroups
containing
A
. More explicitly,
h
A
i
consists of all ±nite products
a
1
···
a
n
,
n
,
2
,...
,
where for each
i
, either
a
i
or
a
−
1
i
belongs to
A
. To see this, note that all such products
belong to any subgroup containing
A
, and the collection of all such products forms a
subgroup. In checking that the inverse of an element of
h
A
i
also belongs to
h
A
i
,weuse
the fact that
(
a
1
a
n
)
−
1
=
a
−
1
n
a
−
1
1
which is veri±ed directly: (
a
1
a
n
)(
a
−
1
n
a
−
1
1
)=1.
1.1.3 Defnitions and Comments
The groups
G
1
and
G
2
are said to be
isomorphic
if there is a bijection
f
:
G
1
→
G
2
that
preserves the group operation, in other words,
f
(
ab
f
(
a
)
f
(
b
). Isomorphic groups are
essentially the same; they di²er only notationally. Here is a simple example. A group
G
is
cyclic
if
G
is generated by a single element:
G
=
h
a
i
. A ±nite cyclic group generated
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '11
 sdd

Click to edit the document details