instructionsMAT 301 Groups and Symmetry – Problem Set 3Due Wednesday, March 23 at 6:10pm
NOTE:
• Students are expected to write up solutions independently. If several students hand in
solutions which are so similar that one or more must have copied from someone else’s
solution, this will be treated as academic misconduct and reported to the department
administration.
• Not all of the questions will be marked. Students will not know in advance which
questions will be marked and are advised to hand in solutions to all of the questions.
• In order to receive full marks for computational questions, all details of the computation
should be included in the solution. Even if the correct final answer is given, marks will
be deducted if some details are left out.
• In solving questions involving proofs, unless the question specifically lists which results
may or may not be used, it is not necessary to reprove facts that have been proved in
class, in the sections of the text that have been covered, or on previous problem sets.
1. Let G be the group GL2(R) of real invertible 2 × 2 matrices under multiplication, and
let H ⊂ G be the subgroup of upper triangular invertible matrices, that is, matrices of
the form

x y
0 z

,
where x, y, z ∈ R, and xz 6= 0. Do left cosets of G with respect to H coincide with
right cosets?
2. Let G be a group. Assume that H is a subgroup of G such that for any a ∈ G, the left
coset of a with respect to H coincides with the right coset: aH = Ha (such subgroups
are called normal). Which of the following statements are true?
(a) For any a ∈ G and any h ∈ H, we have ah = ha.
(b) For any a ∈ G and any h1 ∈ H, there exists h2 ∈ H such that ah1 = h2a.
3. Let G ⊂ Sn be a subgroup. Define a relation ∼ on the set K = {1, . . . , n} as follows.
For i, j ∈ K, say that i ∼ j if j belongs to the orbit orbG(i) of i under the action of
G. Prove that ∼ is an equivalence relation.
4. Prove that the group G1 from Problem 2 of the midterm is isomorphic to the dihedral
group D4 (a complete Cayley table for G1 may be found in solutions).
Hint: To construct an isomorphism between these groups, it may be useful to
look at cyclic subgroups of order 4.
5. The diagram below shows all subgroups for the group D4 of symmetries of a square.
Boxes correspond to subgroups, while an arrow pointing from a box corresponding to
a subgroup H to a box corresponding to a subgroup K means that H ⊂ K.
∗
Given that the box marked by ∗ corresponds to the subgroup {identity map, rotation
by 180◦
, reflections with respect to diagonals}, find the subgroups corresponding to all
other boxes.
Hint: Use Lagrange’s theorem to find possible orders of subgroups of D4. Also,
recall that any group G of order 4 is either cyclic, or isomorphic to Z
∗
8
. How many
subgroups of order 2 does G have in each of these cases?
6. Let G ⊂ S7 be a subgroup of order 8. Prove that there exists i ∈ {1, . . . , 7} such that
for any σ ∈ G we have σ(i) = i.
Hint: Use the orbit-stabilizer theorem to find possible sizes of orbits of G. What
does the result of Problem 3 tell you about these orbits?