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Fearless Symmetry by Robert Gross, Avner Ash

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130 CHAPTER 11
DEFINITION: GL(n, R) is the group of all n-by-n invertible
R-matrices, where the group law is given by matrix
multiplication.
For example, the six matrices (11.2) form the group GL(2, F
2
).
Another thing to remember about matrix multiplication is that
it usually is not commutative. For example, in GL(2, Q)wehave
12
34

56
78
=
19 22
43 50
and
56
78

12
34
=
23 34
31 46
.
Another example drawn from GL(2, F
2
):
11
01

01
10
=
11
10
and
01
10

11
01
=
01
11
.
The Group GL(2 , Z)
The group GL(2, Z) is one way in which number theory and group
theory can come together, though we can give only a few details in
this book. For example, the theory of modular forms, referred to in
chapters 21–23, depends crucially on GL(2, Z) and its properties.
In this section, we will see that even the simple problem of enu-
merating the elements of GL(2, Z) depends on elementary number
theory.
We know that GL(2, Z) is the set of 2-by-2 matrices with integer
entries and determinant equal to 1 or 1. (This is because 1 and 1
are the only elements in Z that have multiplicative inverses in Z.)
It is not that hard to find elements of GL(2, Z), after you know that
the determinant of A =
ab
cd
is ad bc. The entries a, b, c, and
d must all be integers, and the only requirement for invertibility
is that ad bc = 1orad bc =−1. We investigate the case where
ad bc = 1. Then A
1
is given by
d b
ca
. For example, we have
35
47
1
=
7 5
43
.
GROUPS OF MATRICES 131
How do we construct matrices in GL(2, Z)? We start by picking
the first row at random: say [6 15]. Then we look for the second
row [cd] to satisfy 6d 15c = 1. Whoops. That can never happen
if c and d are integers, because the left-hand side is an exact
multiple of 3, and so it could never equal 1. We had better make sure
that a and b share no common factor. Two numbers like that are
called relatively prime. For example, 3 and 20 are relatively prime,
100 and 57 are relatively prime, but 37 and 111 are not relatively
prime, because both are multiples of 37. (Yes, a number is always
considered to be a multiple of itself.)
Armed with this knowledge, we now pick our first row so the two
entries in it are relatively prime: say [6 11]. Now we look for [cd]
to satisfy 6d 11c = 1. One way to do this is to list a lot of multiples
of 6 and 11 and look for entries in the list that are only one
apart:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, ...
11, 22, 33, 44, 55, 66, 77, 88, 99, 110, ...
We immediately notice the 11 on the bottom row and the 12
on the top row, which leads us to choose c = 1 and d = 2. Check:
6 × 2 11 × 1 = 1. We also see 78 on the top list and 77 on the
bottom, leading to the choice of c = 7 and d = 13. Check: 6 × 13
11 × 7 = 78 77 = 1. If we are clever, we can even exploit the 55
on the bottom row and 54 on the top row. The problem is that
54 55 =−1, not 1. But we can solve this by multiplying through
by 1: in fact 54 (55) = 1. This leads to the choice c =−5 and
d =−
9. Check: 6 × (9) 11 × (5) =−54 (55) = 1. We thus
get the following matrices in GL(2, Z):
611
12
,
611
713
,
611
5 9
.
We see that not only have we succeeded, but we have also found
several solutions to our problem. This illustrates a general theorem:
If a and b are relatively prime integers, then there are infinitely
many pairs of integers c and d with the property ad bc = 1.

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