Mathematics 113: Homework # 2
Curtis Tekell
Dr. Poirier
September 12, 2010
Exercise 1.
Prove the following are or are not groups:
(a)
G
=
h
Z
,
+
i
.
(b)
G
=
h
Z
,
×i
.
(c)
G
=
h
Z
/n
Z
,
×i
for some
n
∈
Z
+
.
(d)
G
=
h
(
Z
/n
Z
)
×
,
×i
for some
n
∈
Z
+
.
(e)
G
=
h
M
n
×
m
(
R
)
,
+
i
.
(f)
G
=
h
M
n
×
m
(
R
)
,
×i
.
(g)
G
=
h
M
n
(
R
)
,
×i
.
(h)
G
=
h
Q
[
√
2]
,
×i
.
(i)
G
=
h
U,
×i
, where
U
=
{
z
∈
C
:
∃
n
∈
Z
\ {
0
} 3
z
n
= 1
}
.
(j)
G
=
h
Q
0
,
*i
, where
Q
0
is the set of rationals in lowest form, and
a
b
*
c
d
=
a
+
c
b
+
d
,
reduced to lowest terms.
Proof.
We assume associativity and commutativity of (standard) addition and
multiplication.
(a) Yes. 0 is the neutral element, and for
n
∈
Z
,
n
+ (

n
) = 0.
(b) No. 0
∈
G
is not invertable.
(c) No. If some
m
∈
G
is not relatively prime to
n
, it will not be invertable.
(d) Yes.
0 is the neutral element, and for
n
∈
Z
,
n
+

n
=
0
(e) Yes. The zero matrix
O
is the neutral element; for
A
∈
G
,
A
+ (

1)
A
=
O
.
(f) No. Not all
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 Summer '11
 gelfand
 Chemistry, Elementary arithmetic, Greatest common divisor, D2n

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