Can't seem to get these Practice Exam Questions, Please Help!

Question 13.

(1) Suppose f : G -> H is a homomorphism, and a is an element G has order n. Show that the

order of f(a) must divide n.

(2) Let G be any finite group. Find all homomorphisms f :G->Z.

(3) Prove that (R \ {0}, ·) is not isomorphic to (R, +).

Question 17

(1) Find the order of A = (−1 −1 ) in GL2(R).

( 1 0 )

(2) Find the order of (2, 8,−1) in (Z/9Z)× × Z/28Z × (Z \ {0}). Justify your answer.

Question 21

(1) Define

f : R \ {0} -> R\ {0} by f(x) = x2 (the group operation is multiplication). Prove that f is a homomorphism and give the kernel. Is f an isomorphism?

(2) Prove that {2k | k 2 Z} is a subgroup of (R \ {0}, ·).