150A Algebra
Monica Vazirani
November 3, 2010
Midterm
Name:
ID:
Section:
1.
(10 points) In parts (a)(b), give a careful definition of the terms in boldface. It is set up in
several cases that you can just complete the sentence.
a.
Let
ϕ
:
G
→
G
0
be a homomorphism. Then its
kernel
, ker
ϕ
=
...
Solution:
{
g
∈
G

ϕ
(
g
) = 1
0
}
b.
The
order
of an element
a
∈
G
is ...
Solution:
The smallest positive integer
m
∈
Z
>
0
such that
a
m
= 1
. If no such
finite
m
exists, we say the order is infinite.
2.
(15 points) Let
G
be a group. Suppose
x
∈
G
has (finite) order
s
. Prove that
x
m
= 1 if and
only if
s
divides
m
.
Solution:
⇐
=
If
s

m
then
m
=
sk
for some
k
∈
Z
. Then
x
m
=
x
sk
= (
x
s
)
k
=
1
k
= 1
.
=
⇒
By the Division Algorithm, we can write
m
=
ks
+
r
with
0
≤
r < s
(and
k, r
∈
Z
). Because
1 =
x
m
=
x
sk
+
r
= (
x
s
)
k
x
r
= 1
x
r
=
x
r
, by the minimality of
s
,
it must be that
r
= 0
. But then
m
=
ks
showing
s

m
.
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ID:
Section:
2
3.
(20 points)
In parts (a)  (d), determine if the given statements are
True
or
False
. If it is
true, say so, and give a
brief
explanation. If it is false, say so, and give a counterexample
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 Fall '11
 Wiley
 Differential Equations, Algebra, Equations, Cyclic group, Coset, d. Let σ, Monica Vazirani

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