THE UNIVERSITY OF SYDNEY
Math2968 Algebra (Advanced)
Semester 2
Tutorial Solutions Week 9
2008
1.
(for general discussion) Let
G
,
H
and
K
be groups. In what sense are
G
×
(
H
×
K
)
,
G
×
H
×
K
and
(
G
×
H
)
×
K
the “same”?
Are
G
×
H
and
H
×
G
the “same” in any meaningful sense?
Is it possible to find a
nontrivial finite group
G
such that
G
∼
=
G
×
G
?
Does your answer to the previous question alter if
G
is allowed to be infinite?
Solution
The groups
G
×
(
H
×
K
),
G
×
H
×
K
and (
G
×
H
)
×
K
are all isomorphic. The mapping (
g,
(
h, k
))
mapsto→
(
g, h, k
) for
g
∈
G
,
h
∈
H
,
k
∈
K
is easily seen to be an isomorphism from the first group to the second,
and there is a similar isomorphism from the second to the third.
The groups
G
×
H
and
H
×
G
are isomorphic under the map (
g, h
)
mapsto→
(
h, g
) for
g
∈
G
and
h
∈
H
.
If
G
is a nontrivial finite group then
2
≤ 
G

<

G

2
=

G
×
G

,
so it is impossible even to find a bijection between
G
and
G
×
G
. However, if
G
is infinite then it is
possible for
G
to be isomorphic to
G
×
G
. For example, let
H
be any nontrivial group and put
G
=
{
(
x
1
, x
2
, x
3
, . . .
)

x
i
∈
H
for
i
= 1
,
2
, . . .
}
with coordinatewise multiplication, which is easily seen to yield a group (a direct product of countably
infinitely many copies of
H
). Then the map
(
x
1
, x
2
, x
3
, . . .
)
mapsto→
((
x
1
, x
3
, x
5
, . . .
)
,
(
x
2
, x
4
, x
6
, . . .
))
for
x
1
, x
2
, x
3
, . . .
∈
C
2
is clearly an isomorphism from
G
to
G
×
G
.
2.
Verify that the direct product of abelian groups is abelian, but the direct product of cyclic groups need
not be cyclic.
Solution
If
G
and
H
be abelian groups and
g
1
, g
2
∈
G
,
h
1
, h
2
∈
H
, then
(
g
1
, h
1
)(
g
2
, h
2
) = (
g
1
g
2
, h
1
h
2
) = (
g
2
g
1
, h
2
h
1
) = (
g
2
, h
2
)(
g
1
, h
1
)
,
which verifies that
G
×
H
is abelian. However a direct product of cyclic groups need not be cyclic. For
example, if
C
2
is a cyclic group of order 2, then
C
2
×
C
2
is not cyclic, since its elements have order 1
or 2, so that there is no possibility of
C
2
×
C
2
being generated by a single element, which would have
to have order 4.
3.
Write down the RankNullity Theorem for linear transformations of vector spaces.
(a) Let
T
:
V
→
W
be a linear transformation between vector spaces
V
and
W
both of the same finite
dimension
n
. Use the RankNullity Theorem to verify that
T
is a vector space isomorphism if and
only if
T
is injective if and only if
T
is surjective.
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 One '09
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 Algebra, Cyclic group, direct product, G1 G2, Cmn Cmn

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