4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

7 Suppose that a E G. For each y in the conjugacy class of a, let G(a y)= {g E G: gag= y} be the set of group elements which conjugate a into y. Prove that G(a y) is a coset of CG(a) 1

3.11 Let (G,) be a group such that x²-e for all x E G. Show that (G,.) is abelian. (Here x2 means xx.)

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= - Verify that o is a homomorphism Find Ker(0) (a,b,c,de R ) Use the Fundamental homomorphism theorem to prove that 45