Definition 4
Let (
G,.) be a (multiplicative) group with neutral element
e.
A nonempty subset
H G is a
subgroup of
G
if the following three statements hold
 e H,
 for every
x, y H we have
x y H,
 for every x H the inverse x^{1} of x belongs to H.
Theorem 3
For every subgroup
H of the additive abelian group
(
, +)
there exists an element
a such that
H is the set
of the multiples of
a, that is
H =
a.
Theorem 4
Let
G be a multiplicative group with neutral element
e.
Let
x G an element and
gr(
x)
the subgroup of
H consisting of all powers of
x (including
e =
x^{0} and
x^{1} the inverse of
x).
Let
(
x) be the order of
gr(
x),
that is the cardinality of
gr(
x).
Then two cases arise
 either gr(x) is infinite and then the powers
of x are pairwise different and thus H is isomorphic to
.
 or gr(x) is finite and we have the following properties

(x) is the smallest integer n such that x^{n} = e,

x^{m} = x^{m'} iff
m m'mod (x),
 H is isomorphic to
/n where
n = (x),

H = {e, x, x^{2},..., x^{n1}} where
n = (x).