Section 6.5
Cyclic groups
Cyclic groups, just like cyclic monoids, are well understood:
Theorem
Let G = <g> be a cyclic group
of size n with generator g.
If n is infinite, then G is isomorphic to
(Z,0,+).
If n is finite, then G is isomorphic to
(Z/nZ,0,+).
The size of a finite group or monoid is often referred to as its order.
A cyclic group of order n is denoted by Cn. If n is finite, then we also use Z/nZ for a cyclic group of order n, as Cn is isomorphic to the additive group Z/nZ.
Definition
If G is a group and g 
G, then the order of g
is the smallest positive integer m with gm =
e. If no positive integer m with
gm = e exists, we say that the order of
g is
infinite.
The order of an element g of a group G is equal to the order of the
subgroup of G generated by g. Both are equal to the size
of the set
{e, g, g2, ...}.