Answer to Question #302684 in Abstract Algebra for Jlove

Question

Which of the following sets form a group with respect to the indicated operation:

a. S={x :x∈Z,x<0}; addition

b. S={5x :x∈Z}; addition

c. S={x :x∈Z, x is odd}; multiplication

d. S={-2,-1, 1, 2}; multiplication

8. Do the following: (i) evaluate all the powers of the given permutation; and (ii) find the order of each element.

(a) 1 2 3 4 5 62 3 4 5 6 1 (c). 1 2 3 4 5 66 4 5 2 1 3

(b) 1 2 3 4 5 6 72 1 3 4 6 5 7

9. Verify if the set S={a,b,c,d,e,f,g,h}, with addition and multiplication defined by the table below, is a ring or not. If it is a ring, find the (a) zero element, and the (b) additive inverse of each element.



10. Show that D'={nu:n∈Z}, where u is the unity of an integral domain D, is a subdomain of D.