Can set Z5 be generated by 2 If yes then generate it If no e

Can set Z_5 be generated by [2]. If yes, then generate it. If no, explain clearly why?

Solution

Z₅ generated with respect \'+\' operation:

2 2 mod 5

2+2 4 mod 5

2+2+2 6 1 mod 5

2+2+2+2 8 3 mod 5

2+2+2+2+2 10 0 mod 5

So all the group elements {0,1,2,3,4} in Z₅ can be generated by 2. That is to say, 2 is a generator for the group Z₅. i.e. generated with respect \'+\' operation:

Z₅^* = Z₅ - {0}.Actually all the elements in Z₅^*, {1,2,3,4} can be generated by 2:

2¹ 2 mod 5

2² 4 mod 5

2³ 8 3 mod 5

2⁴ 16 1 mod 5

Z₅^* generated with respect \'.\' operation: