Definition of infinite cyclic subgroup? Let $B$ be a subset of an (additive) abelian group $F$. Then $F$ is free
abelian with basis $B$ iff the cyclic subgroup $\langle b \rangle$ is infinite cyclic for each $b \in B$ and $F = \sum_{b \in B} \langle b \rangle$.
What does it mean for $\langle b \rangle$ to be infinite cyclic?  What's the exact definition?
 A: A group $G$ is cyclic if there exists an element $g_0 \in G$ such that every element of $G$ can be written as a power (positive or negative) of $g_0$. An infinite cyclic group is just what it sounds like: a cyclic group that's infinite. This can be equivalently characterized as follows: a cyclic group $G$ is infinite iff no element of $G$ (besides the identity) has finite order.
It's worth noting a few facts about $G$.


*

*There exists a canonical projection $\pi : \mathbb{Z} \to G$ for every cyclic $G$, where $\pi(1) = g_0$. The projection is injective iff $G$ is infinite, so the corollary is that the only infinite cyclic group is $\mathbb{Z}$.

*If $G, H$ are cyclic groups and $|G| = |H|$, then $G$ and $H$ are isomorphic (consider the isomorphism $f(g_0) = h_0$). Combined with the previous fact, this means that cyclic groups are characterized completely by their order.

*Every subgroup of a cyclic group is also cyclic.

*Any group $\left< b \right>$ is cyclic by definition.
Now we can identify several equivalent conditions to "$\left< b \right>$ is infinite cyclic".
I) $\left< b \right>$ is isomorphic to $\mathbb{Z}$.
II) The map $f : \left< b \right> \to \mathbb{Z}$ given by $f(b^n) = n$ is well-defined (i.e. if $b^n = b^m$, then $n = m$).
III) There exists an onto homomorphism $\left< b \right> \to \mathbb{Z}$.
IV) There exists an imbedding of $\mathbb{Z}$ into $\left< b \right>$.
V) There exist infinitely many imbeddings of $\mathbb{Z}$ into $\left< b \right>$ (note that $f(n) = b^{na}$ is a homomorphism for all $a \in \mathbb{Z}$).
VI) $\left< b \right>$ is infinite.
You could go on, but there are many ways to check that your cyclic group is infinite.
