Homework 2 Solutions
1. Consider the additive group
Z
with the operation +. Which of the following are subgroups
of this group?
1.
{
n
∈
Z
:
n
is prime
}
.
2.
{
0
}
.
3.
{
2
}
.
4.
{
3
n
:
n
∈
Z
}
Solution:
Recall that to check if a subset of a group forms a subgroup, we must check closure,
inverses, and identity. For (i) observe that 2 + 2 = 4 so closure is violated and this is not a
group. For (ii) we have that
{
0
}
is a group. Indeed, it is the trivial group with one element.
It is immediate that closure holds, that 0 is the identity, and that the inverse of 0 is 0. For
(iii) we have the same contradiction as in (i), namely 2+2 = 4 so closure is violated. Finally,
we observe that the set
{
3
n
:
n
∈
Z
}
is a subgroup. Closure follows from the observation
that the sum of two multiples of 3 is a multiple of 3, i.e. if
n
= 3
a
and
m
= 3
b
then
n
+
m
= 3
a
+ 3
b
= 3(
a
+
b
). Since 0 is a multiple of 3 we have the identity, and if
m
= 3
a
then the inverse of
m
, namely

m
=

3
a
= 3(

a
) is also a multiple of 3.
2. Say that a transformation
f
of
R
n
is a
collineation
if for every line
‘
we have that
f
(
‘
) is
a line. Prove that if
f
is a collineation, then
f

1
is also a collineation.
Solution:
Let
‘
⊆
R
n
be a line and choose two points
~x,~
y
∈
‘
. Now, set
~x
0
=
f

1
(
~x
) and
~
y
0
=
f

1
(
~
y
) and let
‘
0
⊆
R
n
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 Spring '11
 RWK
 A2 A3, cycle representation

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