I have a naive question on group theory, which I'm not familiar with in any detail.
Let $H<G$ be a subgroup of some group $G$. Let $g\in G$, $g\not\in H$, but $g^n\in H$ for some $n\geq 2$.
Does there exist a notion of a cyclic subgroup generated by $g$, i.e. $\{g^k|k=0,\dots, n-1\}$, in which $g^n$ is identified with the identity?
The element $g$ is not necessarily a torsion element of $G$.
If $H$ is normal, we can speak of $g$ in the quotient group $G/H$ having order $n$.
Or, we can speak of $G$ acting by left-multiplication on the coset space $G/H$, giving us a homomorphism $G\to\mathrm{Perm}(G/H)$ (the kernel is the normal core of $H$). Then we can speak of $g$ in this image, and describe its cycles. One of its cycles is $(e~g~\cdots~g^{n-1})$, but other cycles may have different lengths. This is not the same as generating a cyclic subgroup of order $n$. We can restrict the action of $\langle g\rangle$ to the cycle $\langle g\rangle H$, in which case the image of the action is literally a group (generated by $g$'s image) of order $n$, but that is artificial.
