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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?

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  • $\begingroup$ If neither $P$ nor $Q$ is normal, then perhaps, $G$ might be a Zappa-Szep product of $P$ and $Q$. $\endgroup$ – Geoffrey Trang Apr 10 at 15:58
  • $\begingroup$ @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. $\endgroup$ – S.D. Apr 10 at 16:02
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The situation where neither $P$ nor $Q$ is normal never occurs. This is because if $q \equiv 1 \pmod p$, then $1<p<q$, so $p \not\equiv 1 \pmod q$.

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