Morphism from Integers to Group

Theorem
Let $$G$$ be a group whose identity is $$e$$.

Let $$g \in G$$.

Let $$\phi: \Z \to G$$ be the mapping defined as:


 * $$\forall n \in \Z: \phi \left({n}\right) = g^n$$.

Then:


 * If $$g$$ has infinite order, then $$\phi$$ is an isomorphism from $$\left({\Z, +}\right)$$ to $$\left \langle{g}\right \rangle$$.
 * If $$g$$ has finite order such that $$\left|{g}\right| = m$$, then $$\phi$$ is an epimorphism from $$\left({\Z, +}\right)$$ to $$\left \langle{g}\right \rangle$$ whose kernel is the principal ideal $$\left({m}\right)$$.
 * Thus $$\left \langle{g}\right \rangle$$ is isomorphic to $$\left({\Z, +}\right)$$, and $$m$$ is the smallest (strictly) positive integer such that $$g^m = e$$.

Proof
By Epimorphism from Integers to a Cyclic Group, $$\phi$$ is an epimorphism from $$\left({\Z, +}\right)$$ onto $$\left \langle{g}\right \rangle$$.

The kernel $$K$$ of $$G$$ is a subgroup of $\left({\Z, +}\right)$.

Therefore by Subgroup of Integers is Ideal and Principal Ideals of Integers, $$\exists m \in \N^*: K = \left({m}\right)$$.

Thus $$\left \langle{g}\right \rangle \cong \left({\Z_m, +}\right)$$.

By Canonical Epimorphism from Integers by Principal Ideal:


 * $$\forall m \in \N^*: \left|{\Z_m}\right| = m$$

So, if $$\left \langle{g}\right \rangle$$ is finite, and if $$\left \langle{g}\right \rangle \cong \left({\Z_m, +}\right)$$, then $$m = \left|{g}\right|$$.

Furthermore, $$m$$ is the smallest (strictly) positive integer such that $$g^m = e$$, since $$m$$ is the smallest (strictly) positive integer in $$\left({m}\right)$$ from Principal Ideals of Integers.

If $$\left \langle{g}\right \rangle$$ is infinite, then $$m = 0$$ and so $$\phi$$ is an isomorphism from $$\left({\Z, +}\right)$$ onto $$\left \langle{g}\right \rangle$$.