Originally Posted by

**tangibleLime** My book defines a generator *a* of a cyclic group as:

$\displaystyle <a> = \left \{ a^n | n \in \mathbb{Z} \right \}$

Almost immediately after, it gives an example with $\displaystyle Z_{18}$, and the generator <2>. but it says...

$\displaystyle <2> = \left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\}$

Where are those numbers (0, 2, 4, ..., 16) coming from? If I follow the definition of a generator exactly,

$\displaystyle <2> = \left \{ 2^n | n \in \mathbb{Z} \right \} = \left \{ \right 2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 \}= \left \{1,2,4,8,16,14,10 \right \}$

And obviously,

$\displaystyle \left\{0, 2, 4, 6, 8, 10, 12, 14, 16\right\} \neq \left \{1,2,4,8,16,14,10 \right \}$

What?