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*.
Fig: 1