Math 541
Solutions to HW #2
1. Let GL
2
(
Z
2
) denote the collection of 2
×
2 matrices with entries in
Z
2
which have
non-zero
determi-
nant.(We listed these matrices out in class.)
(a) Make a multiplication table for GL
2
(
Z
2
).
•
Let A =
±
1 0
0 1
²
, B =
±
1 1
1 0
²
, C =
±
0 1
1 1
²
, D =
±
0 1
1 0
²
, E =
±
1 0
1 1
²
, and F =
±
1 1
0 1
²
.
•
The multiplication table may be expressed as follows:
A
B
C
D
E
F
A
A
B
C
D
E
F
B
B
C
A
F
D
E
C
C
A
B
E
F
D
D
D
E
F
A
B
C
E
E
F
D
C
A
B
F
F
D
E
B
C
A
(b) Which pairs of matrices satisfy
a
·
b
=
b
·
a
?
•
For all
b
∈
GL
2
(
Z
2
),
A
·
b
=
b
·
A
. That is, every element commutes with the identity.
•
Every element also commutes with itself.
•
The only other pair that commutes is (
B,C
).
(c) Are there any elements which commute with
every
other matrix? That is, ﬁnd all elements
a
in
GL
2
(
Z
2
) such that
a
·
b
=
b
·
a
for every
b
in GL
2
(
Z
2
).
•
The only element which commutes with every other element in this table is the identity.
(d) For each matrix
a
, compute
a
,
a
2
,
a
3
, and so on until the pattern is clear. Determine the length
of the repeating cycle for each matrix.
•
Let a = A. Then we have A, A, A,.
... Cycle length is 1.
•
Let a = B. Then we have B, C, A, B, C,.
... Cycle length is 3.
•
Let a = C. Then we have C, B, A, C, B,.
... Cycle length is 3.
•
Let a = D. Then we have D, A, D, A,.
... Cycle length is 2.
•
Let a = E. Then we have E, A, E, A,.
... Cycle length is 2.
•
Let a = F. Then we have F, A, F, A,.
... Cycle length is 2.
2. Consider the group
D
3
.
•
Note
: Recall from class we labeled the three reﬂections
F
T
,
F
R
and
F
L
based on ﬂipping over a
line through the Top vertex, Right vertex or Left vertex of the triangle.
(a) Which pairs of elements of
D
3
satisfy
a
·
b
=
b
·
a
?
•
Every element commutes with itself and with the identity,
e
.
•
The only other pair that commutes is (
R
120
,R
240
), the 120- and 240-degree rotations.