# Possible structure of group of order $pq$ where $p$ and $q$ are distinct primes *but* neither $P$ nor $Q$ is normal in $G$

I have a some confusion regarding the accepted answer solution here: Structure of groups of order $pq$, where $p,q$ are distinct primes.

Consider a group $$G$$ with order $$pq$$ where $$p$$ and $$q$$ are distinct primes. Let $$P$$ be a subgroup of $$G$$ with order $$p$$ and $$Q$$ be a subgroup with order $$q$$ (from Sylow's theorems we know these exist cf. this). The assumption $$q \pmod p \neq 1$$ implies $$P$$ is normal in $$G$$ and $$p \pmod q \neq 1$$ implies that $$Q$$ is normal in $$G$$. We need at least one of these assumptions to write $$G$$ as a semi-direct product of $$P$$ and $$Q$$. The accepted solution to the other questions does that.

But what happens if neither of these assumptions hold true, and neither $$P$$ nor $$Q$$ is normal in $$G$$? Then we can no longer write $$G\cong C_p\rtimes C_q$$ or $$G\cong C_q\rtimes C_p$$ (because for writing a group as a semi-direct product of its subgroups, at least one of the subgroups must be normal cf. Wikipedia). How would the group structure look in that case and how to write a mathematical description of it? The accepted answer does not deal that case and I'm not sure if I'm missing something.

Perhaps at least one of the subgroups $$P$$ or $$Q$$ simply has to be normal? But why?

• If neither $P$ nor $Q$ is normal, then perhaps, $G$ might be a Zappa-Szep product of $P$ and $Q$. – Geoffrey Trang Apr 10 at 15:58
• @GeoffreyTrang Maybe, but I think it's more probable that in this context the subgroup $P$ (s.t. $p > q$) is necessarily normal as hinted by this answer. I just don't know how to prove it. – S.D. Apr 10 at 16:02

The situation where neither $$P$$ nor $$Q$$ is normal never occurs. This is because if $$q \equiv 1 \pmod p$$, then $$1, so $$p \not\equiv 1 \pmod q$$.