niversity of Toronto
Faculty of Arts and Sciences
MAT301H1 - S: Groups and Symmetries
Winter 2024
Homework 1
1 Problems to be submitted
Make sure you follow all the indications as stated in the syllabus.
Don＊t let yourself be impressed by the length of the homework or how the problems look. They
are divided into parts to guide you through the problems and make the ideas accessible.
In this problem set we explore the notion of symmetries of some objects and in the techniques
of counting how many symmetries there are via linear algebra.
In problem 1 we develop the cycle notation and the product of transposition. We prove the sign
is well defined.
Problem 2 explores the dihedral group and it is solved in exactly the same way as we studied
the symmetries of the coloured cube in the class. Think of that example while solving.
Problem 3 develops the notion of order in groups and how to use the ideas we develop in problem
2 to study the orders in the dihedral group.
Problem 4 explores the notion of groups being the same or different (which later we will denote
by the term isomorphic).
Remember, our objective in this course is in great part how to use groups to compute. We
develop the notion of §what do we mean by computing with groups§ by exploring this examples,
which as we move forward will build up towards very beautiful ideas.
1. In lecture we have discussed the symmetric group (i.e. the group of permutations of n distinct elements).
We defined the cycle notation for its elements. We denoted this group by Sn.
(a) (3 points) Write in disjoint cycles all the elements of S4 (that is, the permutations of the 4 elements
1, 2, 3, 4.)
Hint: Don＊t over think it. Just do it.
(b) (1 point) Let a1, ..., ak be different numbers from 1, 2, ..., n. Prove that
(a1, ..., ak) = (ak, ak?1)....(ak, a1)
For example, (1, 2, 3, 4) = (4, 3)(4, 2)(4, 1).
Note: The above parenthesis denote cycle notations of permutations.
(c) (1 point) Use the previous fact to prove the following: All permutations can be written as a com-
position of transpositions.
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(d) (4 points) It is not true that a permutation can be written as a composition of transpositions in a
unique way. However, the following fact is true: The parity of the number of transpositions used
to write a fixed permutation is invariant.
Prove the above fact.
Hint: Taking a smaller than b, what is the parity of the number of times ways that a and b cross
each other as you read the transpositions in a given composition from the original order to the new
one?
(e) (1 point) Given a permutation 考 ﹋ Sn, we define its sign, and denote if by sgn(考), as 1 or ?1
according to whether we used an even number or an odd number of transpositions to write 考.
Explain why the sign is well defined and verify that
sgn(考1考2) = sgn(考1)sgn(考2),
where 考1, 考2 are any two permutations.
2. Let n be a positive integer. Consider the points
Pk = (cos(2羽k/n), sin(2羽k/n)),
for k = 0, 1, ..., n? 1. They are the n vertices of a regular n?gon. Notice that P0 = (1, 0).
We also call the midpoint between Pk and Pk+1 by Mk (the subindices run modulo n, so Mn?1 is the
midpoint between Pn?1 and P0.)
Notice that the vertices and the midpoints alternate. If we read them counterclockwise they are
P0,M0, P1,M1, ..., Pn?1,Mn?1.
They create 2n segments, each with one vertex and one midpoint as endpoints. We will call these
segments by chambers and denote chamber P0M0 by C.
We want to answer the question: what is the structure of the linear transformations 朴 : R2 ?↙ R2
that send the vertices to the vertices and preserve their adjacency. In other words, the isometries of the
n? gon.?
The isometries of the n?gon form a group which is called the Dihedral Group and denoted by Dn.
(a) (1 point) Verify that the isometries of the n?gon always send C to some chamber.
(b) (2 points) Let 朴1,朴2 : R2 ?↙ R2 be any two isometries of the n?gon. Prove that if 朴1 and 朴2
send the chamber C to the same place, then
朴1 = 朴2.
Notice that the above equality is an equality of linear transformations.
(c) (1 point) Let S : R2 ?↙ R2 be the reflection in the X-axis and R : R2 ?↙ R2 a counterclockwise
rotation by an angle of 2羽/n. Justify that S and T are isometries of the n? gon.
(d) (2 points) By tracking its action on C, justify the following fact: all isometries of the n?gon can
be written as compositions of S and R.
(e) (1 point) Use the above facts to justify the following fact: there are as exactly the same number of
chambers than of isometries of the n?gon.
(f) (1 point) How many isometries does the n-gon has?
(g) (2 points) Suppose we ignore the midpoints and we instead call the sides of the n?gon the chambers.
In this case, there are n chambers and so the number of isometries is not the same a the number of
chambers.
What part of the previous process, when we consider the midpoints, breaks down if we only consider
the sides as the chambers? Explain carefully.
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3. We say a transformation has order n if n is the smallest positive integer such that when you perform
the transformation n times you obtain the identity transformation.
For example, in the permutation group S4, the permutation (1, 2, 3) has order 3 because:
(1, 2, 3)(1, 2, 3) = (1, 3, 2)
and
(1, 2, 3)(1, 2, 3)(1, 2, 3) = (1, 3, 2)(1, 2, 3) = (1)(2)(3) = identity.
Another example: for the isometries of the coloured cube, each of the reflections has order 2.
(a) (3 points) Compute the order of all the elements of S4.
Hint: This is easy. Just do it by hand and notice there is a lot of repetition! Don＊t over think it!
(b) (2 points) Prove that in Dn, the dihedral group defined in the previous problem, R has order n.
(c) (1 point) Let 朴 be an element of the Dihedral group Dn. Prove that it has a finite order, that is,
prove there exists a positive integer m such that
朴m = identity.
Hint: What happens with the chambers as you apply succesive powers of 朴? Problem 2(b) will be
useful here.
(d) (2 points) Let 朴 be an element of the Dihedral group Dn whose order is m. Denote by Ck = 朴k(C)
for k = 0, 1, ...,m ? 1. That is, C0, ..., Cm?1 is the set of chambers that you can reach with the
powers of 朴.
Let D be a chamber different from C0, ..., Cm?1, in case there exists one. There exists 朵, an isometry,
such that 朵(C) = D by problem 2. Denote by Dk = 朴k(D) for k = 0, 1, ...,m? 1.
Prove that D0, ...,Dm?1 are different among themselves and different from all of C0, ..., Cm?1.
(e) (2 points) Prove that the order of each element of the Dihedral group divides 2n.
Hint: Using the previous part you can divide the set of all chambers, which has cardinality 2n,
into subsets of order m. These subsets consist of the chambers reached from a given chamber using
朴0, ...,朴m?1.
4. We have constructed several groups so far. Amopng then are the following groups:
1. The permutation group S4,
2. The isometries of the coloured cube (what we did in lecture),
3. The Dihedral group D12, that is, the isometries of the 12?gon (called a regular dodecagon).
We have proven all of them have 24 elements! (Make sure you understand this) In this question we
explore whether they are the same or different group.
A cube has 8 vertices. If v is a vertex, then ?v is also a vertex (the vertex farthest away!). We can
group the vertices into four pairs of antipodal vertices like this.
(a) (2 points) Prove that every isometry of the coloured cube sends a pair antipodal vertices to another
(possibly different) set of antipodal vertices.
Hint: You do not need to check this for the 24 isometries. We have seen in lecture all the isometries
are built out of three specific ones! Use that to your advantage.
(b) (3 points) Call L1, L2, L3, L4 the four pairs of antipodal points we have defined. By the previous
part, there exists a permutation 考 of S4 such that
朴(Li) = L考(i).
(Make sure you understand how to construct it!) Call that permutation by P (朴).
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Justify that
P (朴1 ? 朴2) = P (朴1) ? P (朴2).
Remark: The ? on the right hand side is the composition of permutations in S4, while the ? in
the left hand side is the composition of linear transformations.
(c) (2 points) Prove that if 朴1,朴2 are two permutations of the coloured cube with P (朴1) = P (朴2)
then 朴1 = 朴2.
Hint: Use Linear Algebra!
(d) (1 point) We have explained in lecture that a group is a set with a multiplication table. Explain
why the above parts proves that the multiplication tables of S4 and of the symmetries of the cube
are the same.
You don＊t have to be extremely precise, as we have not developed the exact terminology for this.
We are understanding the notions at the moment.
Remark: Once we define all appropriately, we shall say that the groups are isomorphic and the
map P is an isomorphism.
(e) (2 points) Prove that the Dihedral group D12 has an essentially different table to the other two
groups, despite having the same number of elements.
Hint: Who would correspond to R?