Math 4124
Wednesday, February 1
Second Homework Solutions
1.
2.1.10(a) on page 48
Prove that if
H
and
K
are subgroups of the group
G
, then so is
H
∩
K
.
Let
e
be the identity of
G
.
Then
e
∈
H
,
K
because
H
and
K
are subgroups of
G
,
consequently
e
∈
H
∩
K
. Next let
x
,
y
∈
H
∩
K
. Then
xy
∈
H
because
H
≤
G
and
xy
∈
K
because
K
≤
G
. Therefore
xy
∈
H
∩
K
. Finally
x

1
∈
H
and
K
because
H
and
K
are subgroups of
G
, and we deduce that
x

1
∈
H
∩
K
. It now follows that
H
∩
K
is
a subgroup of
G
as required.
2.
1.2.3 on page 27
Use the standard generators and relations to show that every ele
ment of
D
2
n
which is not a power of
r
has order 2. Deduce that
D
2
n
is generated by
the two elements
s
and
sr
, both of which have order 2.
Each element of
D
2
n
can be written uniquely in the form
r
i
or
sr
i
, where 0
≤
i
≤
n

1
(see page 25). The elements
r
i
are of course powers of
r
, so we need to prove that
sr
i
has order 2. Since
sr
i
=
e
, it will be sufficient to show that
(
sr
i
)
2
=
e
; we shall show
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 Spring '08
 Staff
 Math, Algebra, Group Theory, Prime number, The Elements, Generating set of a group

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