Math 212, Assignment 4

All questions are equally weighted. They will be marked for correctness

and clarity of explanation.

1. Let A be a cube and let G be the group of rotations that are symmetries of the

cube. (Do not include any reflections.) For each point x given below, describe

the set G(x) and the subgroup Gx. Find #G(x) and #Gx in each case, where

#X denotes the number of elements in the set X.

(a) a corner of the cube.

(b) the mid-point of an edge of the cube.

(c) the centre of a face of the cube.

2. Let L be the line

L =

x

1

| x ∈ R

and let G be the ax + b-group which we view as a subgroup of SL as in 3.8.13.

Let

x =

2

1

.

Find G(x) and Gx.

3. Kevin is making beaded children’s bracelets to sell at the community market.

He has an unlimited supply of purple, blue, green and yellow beads. If each

bracelet will have 10 beads, how many bracelet designs are possible?

4. Let G, H, and K be groups and let φ : G → H and ψ : H → K be isomorphisms.

(a) Show that φ

−1 and ψ ◦ φ are both isomorphisms. (The first part of this

amounts to proving Theorem 3.12.3.)

(b) Using (a), show that the isomorphism of groups determines an equivalence

relation on the set of all groups. That is, show that ∼= is an equivalence

relation on the set of all groups.

5. Let G and H be groups and let f : G → H be an isomorphism. Prove that G

is abelian if and only if H is abelian. (This is part 7 of Theorem 3.12.2.)

6. Let G be the set of matrices of the form

(−1)j k

0 1

,

1

where j ∈ {0, 1} and all entries of the matrix are elements of Zn (so all operations on the entries are done modulo n); that is, the entries of the matrix are

really [−1]j

n

, [k]n, [0]n and [1]n. With matrix multiplication, G is a group (you

do not need to prove this). Prove that G is isomorphic to Dn by finding an

explicit isomorphism (and proving that it is an isomorphism).

7. Which of the following groups are isomorphic?

• Z (with addition)

• The subgroup hr

6

i of D8 (with composition)

• Z2 ⊕ Z2 (with componentwise addition)

• 8Z = {. . . , −16, −8, 0, 8, 16, . . . } (with addition)

• Z4 (with addition)

• G = {fn|n ∈ Z}, where fn is the function on Z defined by fn(x) = n + x

for all x ∈ Z (with composition)

• U5 (with multiplication)

• U8 (with multiplication)

By problem 4(b), ∼= is an equivalence relation on the set of all groups, so this

question amounts to sorting the above groups into their equivalence classes.

Each time you add a new group to an equivalence class, give a specific isomorphism from that group to one of the groups already in the equivalence class.

(You must be sure that your function actually is an isomorphism, but you do

not need to include the proof in your answer.) Each time you add a new equivalence class to your list, give a reason why the groups in the new class do not

belong in any of the equivalence classes you have already listed.

8. Let R

× be the set of nonzero real numbers with multiplication. Following the

proof of Cayley’s Theorem, R

× is isomorphic to the subgroup

H = {fg | g ∈ R

×}

of SR× .

(a) Describe the element f10 of H, and the inverse of this element.

(b) Find an element of SR× which is not in H.

(c) Without using your answer from (b), give another reason why H cannot be

all of SR× . (If it is, then SR× is isomorphic to R

×. Using group properties,

explain why this cannot be the case.)

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Rules for group assignments. Make sure you follow the universal rules for group

assignments (below) and any additional rules/procedures laid out in your Group

Contract.

1. Each group member is expected to contribute to the best of their ability, and

assignment submissions should only include the names of group members who

meet this expectation.

2. Each group member should be able to explain the group’s solution to me and

answer any questions I may have about it. It is the whole group’s responsibility

to ensure that this standard is met.

3. The task of composing final solutions and writing them up in good copy must be

shared equally among all group members (after a collaborative problem-solving

process).

4. After good copy solutions are complete, they should be shared among all group

members to be double-checked and proofread. This should be done in advance

of the due date, to allow time for any necessary corrections. Corrections should

be completed by the person who wrote the original solution.

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