Math 4124
Wednesday, January 20
January 20, Ungraded Homework
Exercise 1.1.5 on page 21
Prove for all
n
>
1 that
Z
/
n
Z
is not a group under multiplica
tion of residue classes.
Z
/
n
Z
=
{
¯
0
,...,
n

1
}
. Suppose
Z
/
n
Z
is a group. Then it must have an identity
e
. We
now have
e
=
e
¯
1
=
¯
1, so the identity is
¯
1. Let
x
be the inverse for
¯
0. Then
e
=
x
¯
0
=
¯
0. We
conclude that
¯
1
=
¯
0, which is a contradiction, unless
n
=
1.
Exercise 1.1.8 on page 22
Let
G
=
{
z
∈
C

z
n
=
1 for some
n
∈
Z
+
}
.
(a) Prove that
G
is a group under multiplication (called the group of
roots of unity
in
C
).
(b) Prove that
G
is not a group under addition.
(a) Note that multiplication deﬁnes a binary operation on
G
, because if
x
,
y
∈
G
, then
x
m
=
1
and
y
n
=
1 for some
m
,
n
∈
Z
+
, and then we have
(
xy
)
mn
=
1, which shows that
xy
∈
G
.
Also multiplication is associative and the identity is 1; note that 1
∈
G
. Finally if
z
∈
G
,
then it has an inverse
z

1
; again note that
z

1
∈
G
because if
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 Spring '08
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 Addition, Multiplication, Ring, Binary operation

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