Dr. Marques Sophie
Office 519
Algebra 1
Spring Semester 2016
marques@cims.nyu.edu
Quiz #2
Justify all your answers completely (Or with a proof or with a counter
example) unless mentioned differently.
Problems:
x
1. Is S = {
∈ R2 : x > 0, y > 0} a subgroup of R2 ? Justify.
y
2. Is (Z/nZ, ⊕) a subgroup of (Z, +)? Justify.
3. Let n be a natural integer. Prove that U = {e
(C× , ·).
2πik
n
: 0 ≤ k < n} is a subgroup of
4. Is the map φ : GL2 (R) → GL2 (R) sending A to A−1 a homomorphism of groups?
Justify.
5. Prove that the map det : GL2 (R) → R× sending A ∈ GL2 (R) to det(A) is an
homomorphism of groups. Is it surjective? Compute the kernel of det. Is det
injective? (Justify all the answers).
1