Overview
A group {% G %} is called a cyclic group, if there is an element {% a %} of {% G %}, such that
{% G = \{a^n | n \in \mathbb{Z}\} %}
The element {% a %} is called the generator of the group.
Power Theorem
Finite Group
If {% G %} is a finite group of order {% n %}, then {% a^i = a^j %} iff
{% n %} divides {% i-j %}
Infinite Group
If {% G %} is an infinite group, then {% a^i = a^j %} iff
{% i=j %}
Lemma
{% a^k = e %} iff {% |a| %} divides {% e %}