DISCRETE MATHEMATICS
W W L CHEN
c
±
W W L Chen, 1991, 2008.
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Chapter 9
GROUPS AND MODULO ARITHMETIC
9.1. Addition Groups of Integers
Example 9.1.1.
Consider the set
Z
5
=
{
0
,
1
,
2
,
3
,
4
}
, together with addition modulo 5. We have the
following addition table:
+
0
1
2
3
4
0
0
1
2
3
4
1
1
2
3
4
0
2
2
3
4
0
1
3
3
4
0
1
2
4
4
0
1
2
3
It is easy to see that the following hold:
(1) For every
x,y
∈
Z
5
, we have
x
+
y
∈
Z
5
.
(2) For every
x,y,z
∈
Z
5
, we have (
x
+
y
) +
z
=
x
+ (
y
+
z
).
(3) For every
x
∈
Z
5
, we have
x
+ 0 = 0 +
x
=
x
.
(4) For every
x
∈
Z
5
, there exists
x
0
∈
Z
5
such that
x
+
x
0
=
x
0
+
x
= 0.
Definition.
A set
G
, together with a binary operation
*
, is said to form a group, denoted by (
G,
*
), if
the following properties are satisﬁed:
(G1) (CLOSURE) For every
x,y
∈
G
, we have
x
*
y
∈
G
.
(G2) (ASSOCIATIVITY) For every
x,y,z
∈
G
, we have (
x
*
y
)
*
z
=
x
*
(
y
*
z
).
(G3) (IDENTITY) There exists
e
∈
G
such that
x
*
e
=
e
*
x
=
x
for every
x
∈
G
.
(G4) (INVERSE) For every
x
∈
G
, there exists an element
x
0
∈
G
such that
x
*
x
0
=
x
0
*
x
=
e
.
Chapter 9 : Groups and Modulo Arithmetic
page 1 of 5