Definition 4
Let
be a (multiplicative) group with neutral element
.
A nonempty subset
is a subgroup of
if the following three statements hold

,
 for every
we have
,
 for every
the inverse
of
belongs to
.
Theorem 3
For every subgroup
of the additive abelian group
there exists an element
such that
is the set
of the multiples of
, that is
.
Theorem 4
Let
be a multiplicative group with neutral element
.
Let
an element and gr(
)
the subgroup of
consisting of all powers of
(including
and
the inverse of
).
Let
be the order of gr(
),
that is the cardinality of gr(
).
Then two cases arise
 either gr(
) is infinite and then the powers
of
are pairwise different and thus
is isomorphic to
.
 or gr(
) is finite and we have the following properties

is the smallest integer
such that
,

iff
,

is isomorphic to
where
,

where
.
Marc Moreno Maza
20080107