The map f : (Z,+,0) -> G given by f(i)
= gi is a morphism of groups.
If there is no positive integer n such that
gn = e, then
f is a bijection.
Otherwise, if n is the minimal positive integer n with
gn = e, then
G is isomorphic to (Z/nZ,+,0).
For any i,j 
Z,
gi + j =
gigj and
g0 = e.
(By a previous remark, we need not check the condition on inverses.)
Z,
gi + j =
gigj and
g0 = e. (By a previous remark, we need not check the condition on inverses.)
Clearly, f is surjective.
Suppose there are distinct positive integers i, j such that gi = gj. Then, for n = i - j, we have gn = gi - j = e. Thus, f is injective as well, and hence an isomorphism.
Suppose there are distinct positive integers i, j such that gi = gj. Then, for n = i - j, we have gn = gi - j = e. Thus, f is injective as well, and hence an isomorphism.
By the choice of n, the elements
gi (i = 0, 1, ... , n - 1),
are all distinct.
Now for any m = qn + r, with q the quotient and
r the remainder of m/n, we have
f(m) = gm =
gqn + r
= gnqgr = gr
= f(r) .
In particular, the map
f* : Z/nZ -> G given by
f(m + Z) = gm is well
defined. It is straightforward to check that f* is
an isomorphism of groups.