# A group of order $pqn$ has a non-abelian subgroup of order $pq$

Original (Flawed) Question

Why is it that if a group is of the order $pqn$ where $p, q$ are distinct primes and $n$ is some integer coprime to $p$ and $q$, then there is a non-abelian subgroup of order $pq$? (I am reading some notes and the author says this without proving it, so I assume it is very elementary.)

Revised Question

Sorry about this confusing question, I think I have misunderstood it. (As @QiaoChuYuan kindly suggested.) It should be saying for a cyclic group of order $pqn$ where $p,q,n$ are as described above, AND $p$ divides $q-1$ then there is a non-abelian subgroup of order $pq$. Does this make sense now? If so could someone please tell me why it is true? Thank you.

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This is clearly false in general (take the cyclic group of order $pqn$). What source is this from? Are you sure you aren't misreading it? – Qiaochu Yuan Feb 5 '12 at 18:25
Can you give a link to the notes? – user641 Feb 5 '12 at 18:50
Even if the original group is non-abelian, this is not true. Take the direct product product of a cyclic group of order $pq$ and an nonabelian group of order $n$. – N. S. Feb 5 '12 at 18:54
@SteveD: unfortunately they are of paper-form and not in english! – User1835639 Feb 5 '12 at 20:08
@User1835639: your edited statement is still clearly false, and the cyclic group is still a counterexample. Can you quote from the relevant section of the notes? I think you are misunderstanding something still. – Qiaochu Yuan Feb 5 '12 at 20:13

Even worse: there is no non-abelian group of order $3 \cdot 5$. – Dylan Moreland Feb 5 '12 at 18:36
To piggyback on Dylan's comment, the dihedral group of order 30 is nonabelian of order $3\cdot5\cdot2$, and has no nonabelian subgroup of order $3\cdot5$. – user641 Feb 5 '12 at 18:50
Yup. It sounds like he is trying to say that for a fixed order there exists a group with such properties (which probably follows from some semidirect product construction -- given the "dividing $q-1$" condition). – Matthew Pancia Feb 5 '12 at 21:27