Cyclic group generator of [1, 2, 3, 4, 5, 6] under modulo 7 multiplication

Question

Find all the generators in the cyclic group [1, 2, 3, 4, 5, 6] under modulo 7 multiplication. I got <1> and <5> as generators. The answer is <3> and <5>. Can somebody please tell why is 3 a generator?

Answer 1

You compute the cyclic subgroups of [1, 2, 3, 4, 5, 6] by computing the powers of each element:

• 1 = {1^1 mod 7 = 1, 1^2 mod 7 = 1, ...}
• 2 = {2^1 mod 7 = 2, 2^2 mod 7 = 4, ...}
• 3 = {3, 2, 6, 4, 5, 1}
• 4 = {4, 2, 1}
• 5 = {5, 4, 6, 2, 3, 1}
• 6 = {6,1}

From that you can see that 3 and 5 are cyclic.