3. (a) Let Z2 = {1, 5, 7, 11, 13, 17, 19, 23} be the multiplicative group of integers modulo

24. Let H = {1, 13}.

i. Prove that H is a normal subgroup of Z₂₁.

ii. Find the right cosets of H.

iii. Construct the Cayley table of Z/H.

iv. Which of the Klein Four group and the cyclic group of order 4 is Z/H

isomorphic to?

(b) Let U

= {(8₂%)/₁

Define the map 8: U → R* x R* by

c€ R₁ac +0} ² = {(1₂1) | DER}.

a, b, c € R, ac #0 and H

a

² (8₂6) = (a, c)

where R* is the group given by the set R* = R\{0} with ordinary multiplication

as the binary operation.

i. Show that U is a subgroup of GL₂(R).

ii. Show that is a homomorphism.

iii. Find the kernel and image of 0, showing your working.

iv. Show that H is a normal subgroup of U.

v. Prove that U/H is isomorphic to R* x R*.