completed in 12 hours

HOMEWORK 4

Due Thursday, May 4, at the beginning of discussion

1. Write the Cayley table for the Dihedral group D8 with 8 elements. Why is D8 not

isomorphic to Z/8Z? 2. Describe the following sets:

• 4Z \ 6Z = {x 2 Z: x 2 4Z and x 2 6Z} • 4Z [ 6Z = {x 2 Z: x 2 4Z or x 2 6Z} • 4Z + 6Z = {h + k : h 2 4Z and k 2 6Z}.

For each of set S, either prove that the set is a subgroup of (Z, +), or disprove the claim by means of a counterexample.

3. Let (G, ?) be a group and let H and K be subgroups of G. Prove or disprove each of the

following statements. (If it is true, give a proof; if it is false, give a counterexample.)

(a) H \ K = {x 2 G: x 2 H and x 2 K} is a subgroup of G. (b) H [ K = {x 2 G: x 2 H or x 2 K} is a subgroup of G. (c) (removed to be added to a later assignment)

(d) Show that if (G, ?) is abelian, then H ? K is a subgroup of G.

(e) If G = Z, H = 4Z and K = 6Z, find s, t 2 N such that H \K = sZ and H +K = tZ. How are s and t related to 4 an 6? In general, for H = mZ and K = nZ, make a conjecture about H \ K and H + K.

4. (postponed to a later assignment)

(a) Compute the list of subgroups of the group Z/45Z and draw the lattice of subgroups. (prove that you really got all the subgroups)

(b) How many subgroups does Z/2nZ have? 5. Let (G, ?) be a group and let G2 = {x 2 G: x ? x = e}.

(a) Find G2 for G = (Z, +), (R � {0}, ·), (Z6, +6) and D6. (You may refer back to their group table.)

(b) Disprove the following theorem: For every group G, G2 is a subgroup of G. (Make

sure you explain why your counterexample disproves the theorem.)

(c) Complete the theorem: If G is , G2 is a subgroup of G. Then prove your

theorem.

(d) (optional) Suppose G Is abelian. Is G3 = {x 2 G: : x ? x ? x = e} a subgroup as well? What about Gn (for n 2 N)?

6. Let GL2(R) be the group of 2⇥2 invertible matrices, with multiplication. (The elements of GL2(R) have real entries and non-zero determinant.) Consider the matrix:

A =

1 1

0 1

! .

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a) Find the cyclic subgroup H of GL2(R) generated by the matrix A:

H = hAi = {Ak : k 2 Z}.

b) Find a familiar group isomorphic to H. Explicitly provide an isomorphism (and

check that the given map is, indeed, an isomorphism).

7. Let (G, ?) and (K, �) be groups. Let �: G ! K be a group homomorphism (not neces- sarily an isomorphism). Prove that

(a) �(eG) = eK, so � maps the identity of G into the identity of K.

(b) Show that for all g 2 G, �(g�1) = [�(g)]�1. That is, � maps the inverse of g into the inverse of �(g).

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