Math 171 abstract algebra fall 2014- assignment 5 consider

Question

Math 171: Abstract Algebra, Fall 2014- Assignment 5

1. Let G be a group. The following subgroups will become critical in our study of groups later in the course.

• Z(G), called the center of G, defined by Z(G) = {g ∈ G | hg = gh for all h ∈ G}. This is the set of elements of G that commute with all elements of G.
• For g ∈ G, we define the centralizer of g in G, denoted C_G(g),defined by C_G(g) = {h ∈ G | hg = gh}. This is the set of elements in G that commute with g.
• For H ⊂ G (not necessarily a subgroup), the normalizer of H, defined by N_G(H) = {g ∈ G | gH = Hg}.

Determine, with justification, which of these subgroups are always normal in G. You may assume, without proof, that they are all subgroups of G.

2. Let G be a group with normal subgroups H and K.

(a) Prove HK ≤ G.

(b) Prove that if gcd(|H|, |K|) = 1, then H × K ≅ HK.

3. Prove or disprove each of the following statements.

(a) If H G and G is abelian then G/H is abelian.

(b) If H G and G/H is abelian then G is abelian.

(c) If H, K G and H ≅ K, then G/H ≅ G/K.

(d) If H, K G and G/H ≅ G/K, then H ≅ K.

4. Let C^× be the non-zero points in the complex plane, which is a group under multiplication. Let φ: C^× → R^× be defined by φ(a + bi) = a² + b². One can check that φ is a group homomorphism. Determine ker(φ) and im(φ). Draw pictures of at least 3 different cosets of ker(φ) in C^×, and use the picture to explain the conclusion of the First Isomorphism Theorem applied to φ.

5. Consider the group G = Z × Z under addition. Let H = ((1, 4), (4, 1))

(a) Determine the orders of the elements H + (1, 1) and H + (1, 3) in G/H.

(b) Describe G/H up to isomorphism. Draw a picture to explain what's going on here.

6. Let G be a group.

(a) If G/Z(G) is cyclic, prove G is abelian.

(b) If N = (x^-1y^-1xy: x, y ∈ G), prove N is a normal subgroup of G, prove G/N is abelian.