Math 212, Assignment 3

All questions are equally weighted. They will be marked for correctness and clarity of explanation.

1. Let

H = {(a, b, c) : a, b, c ∈ Z, a + b + c = 0}.

Prove that H is a subgroup of Z

3

.

2. Exercise 3.7.16, Part 1.

3. Let G be a group and let H1 and H2 be subgroups of G.

(a) Prove that H1 ∩ H2 is a subgroup of G.

(b) Suppose that G is finite and H1 and H2 have orders p and q, respectively, where p and q are distinct primes. Prove that H1 ∩ H2 = {e}.

4. Exercise 3.8.10

5. Let θ be a real number and consider

A =

cos(θ) sin(θ)

− sin(θ) cos(θ)

(a) Verify that A is in O2(R).

(b) Using induction, prove that

A

n =

cos(nθ) sin(nθ)

− sin(nθ) cos(nθ)

for all n = 1, 2, 3, . . ..

(c) For which values of θ does A have finite order in O2(R)?

6. Let G = D6 and H =< r2 >= {e, r2

, r4} (which is a subgroup – you should

be able to prove this, but you do not need to include the proof on your

assignment). List all of the left cosets of H in G. Find [G : H].

1

7. Let G = GL2(R) and H = SL2(R). Prove that, for any a, b in G, the cosets

aH and bH are equal if and only if det(a) = det(b).

8. (a) Let G be a cyclic group of order n. Prove that G has a subgroup of

order k for every positive divisor k of n. (In other words, prove that

the converse of Corollary 3.9.12 holds for cyclic groups.)

(b) What are the possible orders of subgroups of A4 (the alternating group

on n elements). Does there actually exist a subgroup of each of these

orders? (You may look up a list of the subgroups of A4.) What

conclusion can you draw about Corollary 3.9.12?

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.

2