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
).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '11
 gelfand
 Chemistry, Morphism, Isomorphism, Det, Group isomorphism, automorphism

Click to edit the document details