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Quiz 2 Practice Problems
Math 332, Spring 2010
Isomorphisms and Automorphisms
1.
Let
C
be the group of complex numbers under the operation of addition, and deﬁne a
function
ϕ
:
C
→
C
by
ϕ
(
a
+
bi
) =
a

bi.
Prove that
ϕ
is an automorphism of
C
.
2.
Let
G
be an abelian group, and deﬁne a function
ϕ
:
G
→
G
by
ϕ
(
a
) =
a

1
. Prove that
ϕ
is an automorphism of
G
.
3.
Let
ϕ
:
Z
10
→
U
(11) be an isomorphism, and suppose that
ϕ
(1) = 8. Determine
ϕ
(3).
4.
Prove that
A
4
is not isomorphic to
D
6
.
5.
List the automorphisms of
Z
8
. Express your answers as permutations of the set
{
0
,
1
,...,
7
}
.
6.
List the four elements of Inn(
D
4
). Express your answers as permutations of the set
{
e,r,r
2
,r
3
,s,rs,r
2
s,r
3
s
}
.
7.
Let
α
∈
Aut(
D
5
), and suppose that
α
(
r
) =
r
2
and
α
(
s
) =
rs
. Find
α
(
r
3
s
).
8.
Let
α
∈
Aut(
Q
8
), and suppose that
α
(
i
) =
k
and
α
(
j
) =

i
. Express
α
as a permutation
of the set
{
1
,

1
,i,

i,j,

j,k,

k
}
.
9.
Determine the isomorphism type of the group whose Cayley table is shown below:
e p q r s t u v
e
e p q r s t u v
p
p r u t q e v s
q
q s r v t u p e
r
r t v e u p s q
s
s v p u r q e t
t
t e s p v r q u
u
u q t s e v r p
v
v u e q p s t r
1
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10.
Let
G
be a group, and deﬁne a function
ϕ
:
G
×
G
×
G
→
G
×
G
×
G
by
ϕ
(
a,b,c
) = (
b,c,a
)
.
Prove that
ϕ
is an automorphism of
G
×
G
×
G
.
11.
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