Math 3560 HW Set 1
Kara
September 3, 2013
We thank Professor Peter Kahn, a previous instructor of this course, and Baik,Kara,Tran,
previous TAs for the course, for allowing us to use their solutions of previous homework
sets whenever it is possible to do it. Appended to this set of solutions are some general
comments that may be helpful to you on future homework sets. Please read them carefully.
(1.3) We adopt the notation of Figure 1.4 in the book. Let’s label the vertices of the
tetrahedron as 1, 2, 3, and 4. If we focus on the vertices, each symmetry of the tetrahedron
is nothing but a permutation of the vertices. First, let’s introduce a notation. If a symmetry
of our labeled tetrahedron (or equivalently, a permutation of the vertices)
maps 1 to 3, 2
1 2 3 4
to 4, 3 to 1, 4 to 2, then we will express the symmetry by
. By using this
3 4 1 2
notation, we see that
1 2 3 4
1 2 3 4
and s =
r=
4 3 2 1
1 3 4 2
Now, we will compute srs and rsrr.
srs = s(rs) =
1 2
=
4 3
1 2
=
3 1
1 2 3
4 3 2
3 4
1
2 1
2
3 4
2 4
1 2 3 4
1 2 3 4
4
4 3 2 1
1 3 4 2
1
2 3 4
4 3 1
Since the vertex 4 remains fixed and other three vertices are mapped to each other, srs is
a rotation about the axis passing through vertex 4.
1
1 2
rsrr = (rs)(rr) =
1 3
1 2 3 4
1
=
2 4 3 1
1
1 2 3 4
=
2 1 4 3
3 4
1 2 3 4
1 2 3 4
1 2 3 4
4 2
4 3 2 1
1 3 4 2
1 3 4 2
2 3 4
4 2 3
Since the vertices 1 and 2 swap their original positions as 3 and 4 swap their original positions, rsrr is a rotation aout the axis passing through the midpoints of edges 12 and 34.
(1.4) We can express the twelve rotational symmetries of the tetrahedron in terms of r and
s.
1 2 3 4
1. r =
, rotation through 2π/3 about the axis passing through vertex 1
1 3 4 2
1 2 3 4
2
, rotation through 4π/3 about the axis passing through vertex 1
2. r =
1 4 2 3
1 2 3 4
3
, identity
3. r =
1 2 3 4
1 2 3 4
, rotation through π about the axis passing through the midpoints
4. s =
4 3 2 1
of edges 14 and 23(notice that s2 =id)
1 2 3 4
, rotation through 2π/3 about the axis passing through vertex 3
5. rs =
2 4 3 1
1 2 3 4
2
, rotation through 4π/3 about the axis passing through vertex 3
6. sr =
4 1 3 2
1 2 3 4
7. sr =
, rotation through 2π/3 about the axis passing through vertex 2
4 2 1 3
1 2 3 4
2
8. r s =
, rotation through 4π/3 about the axis passing through vertex 2
3 2 4 1
1 2 3 4
9. srs =
, rotation through 2π/3 about the axis passing through vertex 4
3 1 2 4
1 2 3 4
10. rsr =
, rotation through 4π/3 about the axis passing through vertex 4
2 3 1 4
2
1 2 3
11.
=
3 4 1
edges 13 and 24
1 2 3
2
12. rsr =
2 1 4
edges 12 and 34
r2 sr
4
, rotation π about the axis passing through the midpoints of
2
4
, rotation π about the axis passing through the midpoints of
3
(1.9) There are 12 vertices on the hexagonal plate. A symmetry has 12 choices for where
to send the first vertex. However, after that choice has been made, there are only 2 choices
for where to send the rest of the vertices, namely deciding if the vertices of each plate will
go clockwise or widdershins. Thus, there are 24 symmetries of the hexagonal plate.
Label the vertices of each hexagon from 0 to 5 consecutively, and label each side of the
plate A and B so that each vertex is named ia or ib , where i ∈ {0, 1, 2, 3, 4, 5}. Then ia is
adjacent to (i−1)a , (i+1)a , and ib (modulo 6). Let r be the rotation of the hexagonal plate
around its center axis by π3 , s be the reflection of the plate across the plane through 1a ,
1b , 4a , and 4b , and t be the reflection of the plate which swaps ia with ib for each i. These
three elements generate 24 distinct elements, hencehitting the entire symmetrygroup. In
particular, the symmetry group consists of the set rk , srk , trk , strk : 0 ≤ k ≤ 5 .
Notice that r6 = s2 = t2 = 1, sr = r−1 s, st = ts, and rt = tr (verify these!). Since t,
r3 , and tr3 commute with everything in the generating set, they commute with the entire
group. On the other hand, since sr = r−1 s (verify this!), it is clear that s and rk for k 6= 0, 3
do not commute with everything, in particular not commuting with r and s respectively.
Furthermore, (srk )r = srk+1 while r(srk ) = sr−1 ri = srk−1 , which implies srk does not
commute with r for any k. Similarly, for all k,
r(strk ) = strk−1 6= strk+1 = (strk )r.
In addition, for k 6= 0, 3,
(trk )s = str−k 6= s(trk ),
(strk )s = tr−k 6= trk = s(strk ).
Thus, the only elements which commute with the entire group are id, r3 , t, and tr3 .
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Some additional general comments.
1. Some synonyms for the word “prove” are: show, demonstrate, argue, explain. Sometimes a question that starts “Why ...?” is asking for a proof.
2. If you have not already done so, read the pages entitled “Homework Expectations”
found on the Homework section of the class website, and try to follow the guidelines.
3. If you have worked with another student in the course on one or more of the homework
problems, this is fine–in fact, we encourage you to do so, if that is the way you like
to work–provided you do the writeup for your homework alone.
4. Many statements that you are asked to prove are phrased (or can be phrased) in the
form “If A then B”. This is known as an implication. Other forms of this implication
are “A implies B” or “A is sufficient for B”. To prove any of these, you should assume
that A is true and then proceed to prove B using this assumption. The statement A
is often called the hypothesis (or antecedent) of the implication; the statement B is
usually called the conclusion (or consequent).
There are a couple of possible pitfalls there that students may fall into. One is to
assume B and then prove A. This amounts to proving B implies A, which is the
converse of the desired implication. An implication may be true while its converse
is not. Another is to first assume A and then, along the way to proving B, actually
assume B in that proof. For obvious reasons, this is known as circular reasoning. Not
good.
5. The author uses the terms “injective” and “surjective”. We also use it in class. They
apply to functions (or mappings or transformations). The first is a synonym for “1-1”
and the second is a synonym for “onto”, both of which appear in calculus and linear
algebra courses. See one of us if these terms are unclear.
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