Algebra 3 2010
Exercises in group theory
February 2010
Exercise 1*:
Discuss the Exercises in the sections 1.11.3 in Chapter I of
the notes.
Exercise 2:
Show that an infinite group
G
has to contain a nontrivial
subgroup, i.e. a subgroup
6
=
G,
{
e
}
.
Exercise 3:
Suppose that
a
2
b
2
= (
ab
)
2
for all
a, b
in the group
G.
Show that
G
is abelian.
Exercise 4*:
Show that
h
(1
,
2
,
3
,
4)
,
(1
,
2)
i
=
S
4
.
Exercise 5:
Let
a
,
b
and
c
be elements in a group
G
.
(1) Show that
a
and
a

1
have the same order.
(2) Show that
ab
and
ba
have the same order.
(3) Show that
abc
and
bca
have the same order. (Try to generalize this to a
general result.)
(4) Find three elements
a, b, c
in the symmetric group
S
3
,
such that
abc
and
bac
have
different
orders.
Exercise 6*:
(1) Let
a
and
b
be commuting elements in
G
of finite orders
m
and
n,
re
spectively. Show that (
ab
)
mn
=
e.
(2) Suppose in addition that
n
and
m
are relatively prime. Prove that if a
power
a
i
of
a
equals a power
b
j
of
b
, then
a
i
=
b
j
=
e
. Use this to prove that
the order of
ab
is
nm.
Exercise 7:
Show that the elements of finite order in an
abelian
group
G
form a subgroup of
G
.
Exercise 8:
Consider the following matrices as elements in the group
GL
(2
,
R
):
1
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a
=
0
1

1

1
b
=
0

1
1
0
Show that
a
has order 3 and that
b
has order 4. Show that the order of
ab
is
infinite.
Exercise 9:
Give examples of the following:
(1) An infinite group
G
with a subgroup
H
6
=
G
and

G
:
H

finite.
(2) An infinite group
G
with a subgroup
H
6
=
{
e
}
and

H

finite.
Exercise 10*:
Suppose that the elements
a, b
in the group
G
satisfy the
relation
aba

1
=
b
2
,
where
b
6
=
e.
(1) Show that
a
5
ba

5
=
b
32
.
(2) Assume that

a

= 5
.
Compute

b

.
Exercise 11:
Define an equivalence relation on the elements of the group
G
by
a
∼
c
b
⇐⇒ h
a
i
=
h
b
i
.
Here
h
a
i
is the cyclic subgroup of
G
generated by
a.
Show that the
∼
c

equivalence class of an element
a
is always finite.
Exercise 12*:
Let
G
be a finite group.
(1) Show that if

G

is even then the number of elements of order 2 in
G
is
odd
. (Consider the mapping
x
7→
x

1
. Which elements are the fixed points
of this map?)
(2) Show that the number of elements of order 3 in
G
is
even
(possibly 0).
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 Winter '11
 PhanThuongCang

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