I'm currently reading a textbook about abstract algebra.
There is a proof that every subgroup of a cyclic group is cyclic.
This proof is using the fact as every proof I have found on the Internet that all cyclic groups have the form $ \langle a\rangle=\{a^n\}$, where $n \in \mathbb{Z}$. But I don't think that this is true, because only cyclic groups under multiplication have this form.
But what is with the other cyclic groups? Doesn't one also have to consider them?
It's just notation.
For each group $(G,\ast)$ (for any binary operation $\ast$ that defines a group on the set $G$), we may write the set $G$ under concatenation (which is the fancy term for putting symbols next to each other and it does not always denote multiplication), via the inclusion map $\iota(g)=g$ because $$\iota(g\ast h)=\iota(g)\iota(h)=gh$$ for arbitrary $g,h\in G$.${}^\dagger$ Powers, multiples, etc., depending on $\ast$, are then simply $g^n$ for arbitrary $g\in G, n\in\Bbb Z$.
This is to save time on writing/typing.
Concatenation may as well be the arbitrary notation to use for an arbitrary group, by fiat. In that case, we write $G$ instead of $(G,\ast)$ when the context is clear.
$\dagger$: Here $\iota$ is known as an isomorphism.