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S
UMMARY OF
9/22 L
ECTURE
We started class by considering the following matrix:
A
:=
±
0
1

1 0
²
We noted that
A
2
=

I
, that
A
4
=
I
, and that
A
∈
SL
2
(
Q
)
(which means that
A
is also in
GL
2
(
Q
)
).
Some of you, in your work on problem 1.5, made the false claim that if
g
n
=
I
for some matrix
g
, then
g
must be the identity; the matrix
A
above provides a counterexample. I also pointed
out that I’d made an error in my proof (given during the 9/10 lecture) that the group
Q
×
is not
cyclic: if
α
were a generator of
Q
×
and
α
n
= 1
, we can only deduce that
α
=
±
1
. (In class I
claimed that
α
would have to equal 1 under the circumstances.) Of course, the proof can still
be made to work, since neither
1
nor

1
generates
Q
×
.
Having deﬁned the matrix
A
, we examined the cyclic group generated by it:
{
1
,A,A
2
,A
3
} ≤
SL
2
(
Q
)
.
This is highly reminiscent of some subgroups we’ve run across before, namely
{
1
,i,i
2
,i
3
} ≤
C
×
and
{
1
,α,α
2
,α
3
} ≤ {
symmetries of the square
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 Fall '10
 GideonMaschler

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