54
2.54
(i)
De
fi
ne the
special linear group
by
SL
(
2
,
R
)
= {
A
∈
GL
(
2
,
R
)
:
det
(
A
)
=
1
}
.
Prove that SL
(
2
,
R
)
is a subgroup of GL
(
2
,
R
)
.
Solution.
It suf
fi
ces to show that if
A
,
B
∈
SL
(
2
,
R
)
, then so is
AB
−
1
; that is, if det
(
A
)
=
1
=
det
(
B
)
, then det
(
AB
−
1
)
=
1.
Since det
(
UV
)
=
det
(
U
)
det
(
V
)
, it follows that
1
=
det
(
E
)
=
det
(
BB
−
1
)
=
det
(
B
)
det
(
B
−
1
).
Hence, det
(
AB
−
1
)
=
det
(
A
)
det
(
B
−
1
)
=
1.
(ii)
Prove that GL
(
2
,
Q
)
is a subgroup of GL
(
2
,
R
)
.
Solution.
Both the product of two matrices with rational entries
and the inverse of a matrix with rational entries have rational en
tries.
2.55
Give an example of two subgroups
H
and
K
of a group
G
whose union
H
∪
K
is not a subgroup of
G
.
Solution.
If
G
=
S
3
,
H
=
(
1 2
)
, and
K
=
(
1 3
)
, then
H
∪
K
is not a
subgroup of
G
, for
(
1 2
)(
1 3
)
=
(
1 3 2
) /
∈
H
∪
K
.
2.56
Let
G
be a
fi
nite group with subgroups
H
and
K
. If
H
≤
K
, prove that
[
G
:
H
] = [
G
:
K
][
K
:
H
]
.
Solution.
If
G
is a
fi
nite group with subgroup
H
, then
[
G
:
H
] = 
G

/

H

.
Hence, if
H
≤
K
, then
[
G
:
K
][
K
:
H
] =
(

G

/

K

)
·
(

K

/

H

)
= 
G

/

H
 = [
G
:
H
]
.
2.57
If
H
and
K
are subgroups of a group
G
and if

H

and

K

are relatively
prime, prove that
H
∩
K
= {
1
}
.
Solution.
By Lagrange
’
s theorem,

H
∩
K

is a divisor of
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 Fall '11
 KeithCornell
 Prime number, GL, Subgroup, Det, Cyclic group

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