Download Homework 9 Due Wednesday, April 7 1. Let G = Z 18 and H = 〈 [6

Homework 9
Due Wednesday, April 7
1. Let G = Z18 and H = h [6] i. What is the order of 5 + H in G/H?
2. (a) Let G be a group that is not abelian. Prove that G/Z(G) is not
cyclic.
(b) Prove that if G is a group with |G| = pq where p and q are prime
numbers, then either G is commutative or Z(G) = {1}.
3. Let C× be the group of nonzero complex numbers under multiplication.
Let S 1 = {z ∈ C× : |z| = 1} be the circle group with the operation of
multiplication. Prove that S 1 / C× and C× /S 1 ∼
= R+ , where R+ denotes
the group of positive real numbers under multiplication.
4. Let G = R× be the group of nonzero real numbers under multiplication.
Let N = {−1, 1}. Prove that N is a normal subgroup of G and that G/N is
isomorphic to R+ , the group of positive real numbers under multiplication.
5. Let G be a group. For each g ∈ G let γg : G → G by γg (x) = gxg −1 . Let
Inn(G) = {γg : g ∈ G}.
(a) Prove that Inn(G) is a group under composition. (We call Inn(G)
the group of inner automorphisms of G.)
(b) Prove that Inn(G) ∼
= G/Z(G).
(c) If D2n is the group of symmetries on the regular n-gon, then how
many elements are in Inn(D2n )?
6. An element a of a group G is said to be a square if a = b2 for some b ∈ G.
Let G be an abelian group and let H be a subgroup of G. If every element
of H is a square, and every element of G/H is a square, then prove that
every element of G is a square.
7. Let G be a finite group and let H / G.
(a) Prove that the order of the element gH in G/H must divide the order
of the element g in G.
(b) Prove that if x ∈ G and gcd(|x|, |G/H|) = 1, then x ∈ H.
8. Prove that
(a) U (Z9 ) ∼
= Z6
(b) U (Z15 ) ∼
= Z4 × Z2