Talk:Group of Order p q is Cyclic

Not so sure I agree with this renaming as the title implies that every group of order $p q$ is cyclic, whereas there are conditions under which you can create such a group and it's not cyclic. Hence my original less precise name for this result, whose intention was just to indicate that this was a result "about" such groups of order $p q$.

We also have the complication of whether / how to merge this with Condition for Nu Function to be 1 which (even though I raised the suggestion) I'm no longer sure about, as it's good to have a separate page for the $2$-group case. --prime mover (talk) 17:40, 17 June 2022 (UTC)

Well, the original name and statement were actually not proving that these are necessary and sufficient conditions for a group of order $pq$ to be cyclic, so that name was also misleading. Additionally, it deviated from our standard naming scheme.

I found the name fine because strictly speaking, it does not say that $p$ and $q$ are primes or anything else, so that it is sensible that a reference is checked. But I appreciate that it is not 100% accurate.

Potentially I could settle for "characterisation of cyclic groups of order $pq$" but I am not a fan. And actually then we are very close to the general result regarding the $\nu$ function you referred to. But I do agree that the case of $2$ groups deserves to remain separate, with one proof following from the general theorem. However, for reminding us of that job I consider the {{mergeto}} to be quite useful. — Lord_Farin (talk) 18:08, 17 June 2022 (UTC)

Yep okay, fair enough, I don't feel strongly about it. --prime mover (talk) 20:37, 17 June 2022 (UTC)