19. 20. 21. 22. 23. 24. 25. Consider the groups U(8) and Z4 c) d) e) f) Let H₁, H₂ and H, be abelian groups. Prove or disprove: H, xH₂x H, is an abelian group. List the elements of U(8) x Z4 Determine the identity element of this group. Determine all the elements which are of order two in this group. Determine all the elements of order 4 in this group.? Determine the subgroup generated by the element (7,1) Let K = {xe G|xg=gx Vge G} Show that K is a normal subgroup of G. Define f: R² R² by f(x,y) = (x + 2y, 0) h) i) a) b) Show that f is a homomorphism from into itself Find Ker(f) Let 0: - be defined by 0 la b C by σ(n) =i" a) b) Prove that is a homomorphism Determine Ker (0) Let G= < C. Define = a+d a: - Verify that o is a homomorphism Find Ker(0) (a,b,c,de R ) Use the Fundamental homomorphism theorem to prove that 45

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