# A group of order 561 is cyclic.

Prove that any group of order 561 is cyclic.

• As you are a new user I would like to tell you that : It would not be enough to just write the question to get a reply... please explain what you have tried? – user87543 Dec 13 '13 at 7:17
• 15 not prim but is cyclic. – ayoob Dec 13 '13 at 7:20
• @Magdiragdag : fine fine :) – user87543 Dec 13 '13 at 7:23
• @ayoob: At least you should describe your background, what did you learn and what difficulties did you encountered so that you know what hints and answers should be given. – user99914 Dec 13 '13 at 7:27
• N(H)/C(H) -----> AUT(H) normalizer - central theorem – ayoob Dec 13 '13 at 7:31

In general, there is only one group of order $n$ iff gcd$(n,\varphi(n))=1$. Of course such a group must be necessarily cyclic. 561 satisfies the condition.

• Isn't this more advanced for a beginner? – user87543 Dec 13 '13 at 8:43
• ????????????????????????? – ayoob Dec 13 '13 at 9:44
• @ayoob - what is your confusion? – Nicky Hekster Dec 13 '13 at 11:58
• @NickyHekster This is really a neat fact. I had never heard of it either, until now. – rschwieb Dec 18 '13 at 16:05
• @rschwieb Yes, I think so too! And the nice thing of it it is so simple and you will never forget it! – Nicky Hekster Dec 18 '13 at 16:26