Let G be a group with the following property: Whenever a, b, and c belong to G and ab = ca, then b = c. Prove that G is Abelian. ("Cross cancellation" implies commutativity.)

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

Suppose that in the group G, (ry)2 = r²y² for all x, y E G. Prove that G is abelian.

Which of the following groups are isomorphic? i) Z/75Z ii) Z/3Z e Z/25Z iii) Z/15Z e Z/5Z iv) Z/3Z e Z/5Z OZ/5Z.

Ity and U 33. Let G be a group and suppose that (ab)² = a²b² for all a and b in G. Prove that G is an abelian group. 34 Find all the suborouns of Za x Z, Use this information to show that Z, x Z, is not the

5.PS-16 Question Help A shoe store uses a 60% markup for all of the shoes it sells. What would be the selling price of a pair of shoes that has a wholesale cost of $55? The selling price would be $ (Round to the nearest cent as needed.) Enter your answer in the answer box and then click Check Answer. All narte chowing Review progress Question 9. of 9 pe here to search

Prove that if G/Z(G) is cyclic, then G is abelian.

On Z is defined the following binary operation; x . y = x + y - 1 Show that (Z, . ) is an abelian group

1. Let G and H be groups, and let a: G H be a homomorphism. Show that if a is bijective, then a¹: HG is also a homomorphism.