Graphics Reference
In-Depth Information
As usual in this case, if g
1
and g
2
belong to G, then we shall write g
1
·g
2
instead of the
functional form · (g
1
,g
2
).
Definition.
The pair (G,·) is called a
group
provided that the operation · satisfies:
(1) (Associativity) For all elements g
1
, g
2
, and g
3
in G,
ggg
◊
(
◊
)
=◊
(
ggg
)
◊
.
1
2
3
1
2
3
(2) (Identity) There is an element e in G, called the
identity
of G, such that
eg ge g
◊=◊=,
for all g in G.
(3) (Inverse) For every g in G, there is an element g
-1
in G, called the
inverse
of
g, such that
-
1
-
1
gg
◊
=◊
g
g e
=
.
B.4.1. Example.
The symmetric group of degree n, S
n
, along with product
operation
, which we discussed in the last section is a finite group with n!
elements.
It is not hard to show that the identity and the inverse of each element in a group
is unique. Also, it is always the case that
-
1
(
)
-
1
g
=
g
.
The simplest group is one that consists only of an identity element.
Definition.
A group consisting only of an identity element is called a
trivial group
.
Definition.
A group (G,·) is said to be an
abelian
or
commutative group
if it also
satisfies:
(4) (Commutativity) For all g
1
and g
2
in G, g
1
·g
2
= g
2
·g
1
.
B.4.2. Example.
The standard examples of abelian groups are
Z
,
Q
,
R
, and
C
with
respect to addition. The sets
Q
- 0,
R
- 0, and
C
- 0 become groups under multipli-
cation. The sets
R
n
and
Z
n
become groups using the
vector addition
(
xx
,
,...,
x
)
+
(
yy
,
,...,
y
)
=+
(
x yx
,
+
y
,...,
x
+
y
)
.
12
n
12
n
1
12
2
n
n
B.4.3. Example.
For each positive integer n let
Z
n
=
{
012
, , ,...,
n
-
1
}
and define an operation +
n
on
Z
n
as follows: If a, b Œ
Z
n
, then set
