1
Math 417 Exam II Solutions: July 24, 2009
1
.
If
α
=
β
1
· · ·
β
t
is a product of disjoint
r
i
cycles
β
i
, prove that
α
has order
m
= lcm
{
r
1
, . . . , r
t
}
.
This is Proposition 2.55(ii).
2
(
i
)
.
(10 points) If
G
is a group with Aut(
G
) =
{
1
}
, prove that
g
2
= 1 for every
g
∈
G
.
If
G
is not abelian, then there is
a
∈
G
which in not in the center
Z
(
G
);
that is, there is some
g
∈
G
with
aga

1
=
g
. Hence,
γ
a
:
G
→
G
, conjugation
by
a
, defined by
x
→
axa

1
, is an automorphism of
G
that is not the identity
1
G
because
aga

1
=
g
.
If
G
is abelian, then we have proved that
ι
:
g
→
g

1
is an automorphism of
G
. Now
ι
= 1
G
if there exists
x
∈
G
with
x
=
x

1
. Hence,
ι
= 1
G
if
g
=
g

1
for all
g
∈
G
; that is, if
g
2
= 1 for all
g
∈
G
.
Remark: Here is a sketch of how to treat this last case.The hypothesis
g
2
= 1
for all
g
∈
G
implies that
G
can be viewed as a vector space with scalars in
I
2
. If

G

>
2, then
G
has a basis
x, y, . . .
. Define an automorphism
ϕ
of
G
by
ϕ
(
x
) =
y
,
ϕ
(
y
) =
x
, and
ϕ
(
z
) =
z
for all other basis elements, if any. Hence, if
Aut(
G
) =
{
1
}
, then

G
 ≤
2.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Math, Algebra, Group Theory, Normal subgroup, Abelian group, Quotient group, Homomorphism

Click to edit the document details