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
nonzero
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 240degree rotations.
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.
 Fall '09
 Pollack
 Algebra, Determinant, Multiplication, Matrices, multiplication table, cycle length

Click to edit the document details