Consider the group made up of al elements different than 0 o

Consider the group made up of al elements different than [0] of the set Z₁₃ under the multiplication operation. Find <[4]> (the susbset generated by [4])

Thanks!

Solution

[4] = The elements are obtained as follow:

4 = 4(mod13)

4.4 = 16 = 3(mod13),

4.4.4= 16.4=64=12mod(13),

4⁴ = 12.4 = 48=9(mod13),

4⁵ = 9.4 = 36=10(mod13),

4⁶ = 10.4 = 40 = 1(mod13),

4⁷ = 1.4 = 4(mod13),

4⁸ = 4.4 = 3(mod13),

Now we will get same elelments as before;

Hence [4] = <4, 3, 12, 9,10,1> or <1,3,4,9,10,12>