A non empty subset H of a group G is a subgroup of G if, under the binary operation defined on G, H is a group.
In order to test whether a subset is a subgroup it is sufficient to complete the following three tests:
then 
then 
For example, the even integers are a subgroup of the integers under addition. Certainly the identity 0 is an even integer, the sum of two even integers is an even integer and the inverse of an even integer is an even integer.
The rotations and the identity in 
are a subgroup of .
However, the reflections in together with the identity do not
constitute a subgroup.
For any group G and 
, let 
be the
subset defined by 
. This is a subgroup of
G and is the cyclic subgroup generated by x.
So, in the non zero real numbers under multiplication the set
is a cyclic subgroup.
A very important result concerning subgroups of finite groups is Lagrange's Theorem which states that the order of a subgroup divides the order of a group.
With regard to finite cyclic groups, if G is a
finite cyclic group of order n and x is a generator, then
is a cyclic subgroup of order 
.
Indeed, every subgroup of a cyclic group is cyclic. Now Lagrange's
Theorem would tell us that a cyclic subgroup of a finite cyclic
group of order n must have order a divisor of n. In this case,
the converse is true, namely, for every divisor m of n there is
a cyclic subgroup of order m. It is generated by 
where
x is a generator of the cyclic group and k satisfies km = n.
This converse statement is not true for groups in general.