Homework 7

Homework 7
Math 332, Spring 2013
These problems must be written up in LATEX, and are due this Friday, March 22.
1. Let G =
1 a a∈Z .
0 1 (a) Prove that G is a subgroup of GL(2, R).
(b) Prove that G is isomorphic to Z.
2. Let G, H, and K be groups, and let φ : G → H and ψ : H → K be isomorphisms.
Prove that ψ ◦ φ is an isomorphism from G to K.
3. Prove that there exists a subgroup of GL(3, R) that is isomorphic to GL(2, R)
4. Let G = U (15), the multiplicative group of units modulo 15.
(a) Find an automorphism σ of G such that σ(2) = 7 and σ(14) = 14.
(b) Find an automorphism ρ of G such that ρ(2) = 7 and ρ(14) = 11.
(c) List the eight automorphisms of G.
(d) What is the isomorphism type of Aut(G)? Explain.