[Math] Conditions for cyclic quotient group

Question

Let $G$ be an arbitrary finite group and $H$ a normal subgroup.

What are some good conditions on $H$ that make the quotient $G/H$ cyclic?

I want to avoid any further restriction on $G$.

Answer 1

This is largely the wrong question. Let $H,K$ be groups and let $G = H \times K$ so that $H \trianglelefteq G$. (This is not the only way to get to the upcoming punchline.)

What sort of conditions can you apply to $H$ to force $K \cong G/H$ cyclic? (Notice that $K$ was chosen entirely independently of $H$.)

There are no such conditions generally. You say you don't want to constrain $G$, but $G$ is the only thing that controls the cosets of $H$. $H$ is stuck in one of its cosets and has no control over the rest of $G$.

Notice that no one is giving you constraints on $H$. They constrain the number of cosets of $H$ (via $|G|/|H|$) or they require $G$ special (see especially the finite version of special). This is because constraining a tiny chunk of $G$ is too weak to control the cosets of that chunk.