Mathematics 113: Homework # 2
Curtis Tekell
Dr. Poirier
September 22, 2010
Exercise 1.
(a) Deﬁne the orthogonal group
O
(
n
) =
{
M
∈
GL
(
n,
R
) :
det
(
M
) =
±
1
}
. Then
G
=
h
O
(
n
)
,
×i
is a group.
(b) Deﬁne the special linear group
SL
(
n,
R
) =
{
M
∈
GL
(
n,
R
) :
det
(
M
) = 1
}
.
Then
G
=
h
SL
(
n,
R
)
,
×i
is a group.
Proof.
(a) We need only show that
O
(
n
)
≤
GL
(
n,
R
). Let
A,B
∈
O
(
n
). Then as
det
(
B
) =
±
1
6
= 0,
B

1
exists, and clearly
AB

1
∈
GL
(
n,
R
). Additionally,
det
(
AB

1
) =
det
(
A
)
det
(
B

1
) =
det
(
A
)
det
(
B
)
=
±
1
±
1
=
±
1
,
thus
AB

1
∈
O
(
n
). It follows
O
(
n
)
≤
GL
(
n,
R
).
(b) We need only show that
SL
(
n,
R
)
≤
GL
(
n,
R
). Let
A,B
∈
SL
(
n,
R
). Then
as
det
(
B
) = 1
6
= 0,
B

1
exists, and clearly
AB

1
∈
GL
(
n,
R
). Additionally,
det
(
AB

1
) =
det
(
A
)
det
(
B

1
) =
det
(
A
)
det
(
B
)
=
1
1
= 1
,
thus
AB

1
∈
SL
(
n,
R
). It follows
SL
(
n,
R
)
≤
GL
(
n,
R
).