List all the elements of Z4 times Z8 that have order 4Soluti

List all the elements of Z_4 times Z_8 that have order 4.

Solution

Solution : Z₄ = { 0,1,2,3,} and Z₈ ={0,1,2,3,4,5,6,7} and the identity element of the set (0,0)

A = Z₄XZ₈= { ( 0,0) , (0,1) , ( 0,2) ------(1,0) ,(1,1) ----- ,(4,8) }

Consider the elements (0,2) , (0,6) , (1,2) ,( 1,6) , ( 2,2) , (2,6) , (3 ,2) , (3,6) ----(1)

if we operaate these elements 4 times we get the identity element (0,0)

for ex consider (2,6) operate 4 times we get ( 8 , 24 ) =(0,0) as 8 =0mod4 and 24=0mod8

hence all elements given in step 1 will have order =4