MATH 236H SPRING 2010
HOMEWORK 4
DUE FEBRUARY 10, 2010
This week not all problems are in the book.
(i) Do problems 21, 46 and 53 of Section 8.
(ii) Do problems 27a, 27b, and 29 of Section 9.
(iii) Rotations of the tetrahedron, cube and octahedron.
These problems concern the groups of rotations of the cube and tetrahedron. Let C denote the cube with vertices at the eight points
{(±1, ±1, ±1)} ,
where the three signs are independent of each other. This cube has six
faces. Their centers are at the six points
{(±1, 0, 0), (0, ±1, 0), ( 0, 0, ±1)} .
These form the vertices of a regular octahedron O. Its eight faces are
centered at the points
1
1
1
± ,± ,±
,
3
3
3
again with independent signs. Any rotation of the cube C is a rotation
of the octahedron O and vice versa, so C and O have the same rotation
groups. We will denote this group by G.
The four vertices of C having an even number (0 or 2) of minus signs are
equidistant from each other and form the vertices of a regular tetrahedron
T . The remaining four vertices of C (the ones with an odd nunber of minus
signs) form the vertices of a second regular tetrahedron T 0 . A rotation of C
may preserve T or it may interchange it with T 0 . Any rotation of T carries
C along with it, so the rotation group of T is a subgroup of G, which we
will denote by H.
Any rotation can be represented by a 3 × 3 matrix with determinant 1.
Such a matrix acts on the set of vertices as follows. Write each vertex as a
column vector and multiply it on the left by the matrix, thereby getting a
column vector corresponding to another vertex.
G consists of the 3 × 3 matrices M satisfying the following conditions:
(a) Each entry of M is 0, 1 or −1.
(b) Each row and column of M has precisely one nonzero entry.
(c) The determinant of M is 1.
The binary operation in G is matrix multiplication.
1
2
HOMEWORK 4
Do the following problems. Show your work and prove your answers.
1. Determine the order of G, i.e., count the matrices satisfying the conditions (a)–(c).
2. In G find
• a matrix α of order 3.
• a matrix β of order 4.
• a nondiagonal matrix γ of order 2.
• a diagonal matrix δ of order 2.
3. Determine how many matrices in G satisfy each of the four conditions
of the previous a problem. The sum of these numbers should be one
less (since we have not listed the identity element) than the order of
the group.
4. Consider the four diagonal lines of the cube C:
• The line L1 passing through the origin and ±(1, 1, 1)
• The line L2 passing through the origin and ±(−1, 1, 1)
• The line L3 passing through the origin and ±(1, −1, 1)
• The line L4 passing through the origin and ±(1, 1, −1)
Describe how each of your four matrices (found in 2) permutes these
four lines.
5. Which of these four permutations is even? Which of your matrices
preserves the tetrahedron T ?
6. The action of G on the four diagonal lines defines a homomorphism φ
from G to S4 . What can you say about it? In proving your answer,
you may assume that each matrix of a given type (as in 2) defines the
same type of permutation as your matrix does.
7. What is the image of H under φ?