Example 1.
This example gives a simple group with two elements.
•
We start with the set of two integers,
G
=
{
0
,
1
}
.
•
DeFne a binary operation, denoted by
⊕
,on
G
as shown by
the Cayley table below:
⊕
01
0
1
10
•
±rom the table, we readily see that
0
⊕
0=0
,
0
⊕
1=1
,
1
⊕
0=1
,
1
⊕
1=0
.
•
It is clear that the set
G
=
{
0
,
1
}
is closed under the operation
⊕
and
⊕
is commutative.
•
To prove that
⊕
is associative, we simply determine
(
a
⊕
b
)
⊕
c
and
a
⊕
(
b
⊕
c
)
for eight possible combinations of
a
,
b
, and
c
with
a
,
b
, and
c
in
G
=
{
0
,
1
}
, and show that
(
a
⊕
b
)
⊕
c
=
a
⊕
(
b
⊕
c
)
,
for each combination of
a
,
b
, and
c
.
•
±rom the table, we see that
0
is the identity element, the
inverse of
0
is itself and the inverse of
1
is also itself.
•
Thus,
G
=
{
0
,
1
}
with operation
⊕
is a commutative group of
order
2
.
•
The binary operation
⊕
deFned on the set
G
=
{
0
,
1
}
by the
above Cayley table is called
modulo-2 addition
.
45