Question: Let a be an element of a group and suppose that a has infinite order. How many generators does $\left \langle a \right \rangle$ have?
Following from the hypothesis that a has infinite order, distinct power on a are distinct group elements.
A bit of a brick wall here.
Only hints are appreciated unless it involves substantial number theory. Thanks in advance.
Suppose $b=a^k$ is a generator of the subgroup $<a>$. Then, as as $a\in<a>$, there must be $l\in\mathbb N^\ast$ such that $b^l=a$.
But then $(a^k)^l=a$, so $a^{kl-1}=e$ ($e$ unit element). What does this say about $a$, or $k$ ?