Reference no: EM131083555
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 CG(g),defined by CG(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 NG(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.
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