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Let $G$ be a group with $ord(G)=319$. Proove that $G$ is a cyclic group.

Answer: $ord(G)=319=11*29=n$, the Euler's totient function gives $\phi(n)=\phi(11*29)=\phi(11)*\phi(29)=10*28$. Since $gcd(319,280)=1$, $G$ is a cyclic group.

[Related:Answer from Nicky Hekster]

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serge
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  • I suggest you edit your answer especially the part . " If and only if ... " I do not understand you that you are trying to use in that step.

    On the other hand , you can reach the result using that : Let $p$ and $q$ be primes such that $p>q$. If $q\equiv 1\mod p$, then every group of order $pq$ is isomorphic to the cyclic group $\Bbb Z/pq\Bbb Z$. (see Algebra by chapter III, proposition 6.1)

     – carlos arturo Hurtado Jul 20 '16 at 17:30
  • As ajotatxe pointed out: This "if and only if" assertion in your answer is false. In fact, there are cyclic groups of any positive integer order. Furthermore, the existence of a cyclic group of order 319 does not imply that any group of that order must be cyclic. – Greg Martin Jul 20 '16 at 17:36
  • It is edited now. – serge Jul 20 '16 at 18:29
  • Why does $\gcd(x, \phi x) = 1 \implies $ groups of order $x$ are cyclic? Is that a theorem? if so which one? – fleablood Jul 20 '16 at 18:54
  • In the link (comment from Nicky Hekster), stands that theorem. – serge Jul 20 '16 at 18:58
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    Yes, but the question is just how "well-known" is it, and is the student asked to prove this expected to be able to cite or prove this result? Citing "Nicky Hekter on the internet says this is a well known result" wouldn't be enough to convince me a student has understood and convinced him/herself that he/she understands why this is true? Can you prove the result on your own? Or can you cite a source where it is proven? Furthermore, does this result have a name? – fleablood Jul 20 '16 at 19:07
  • Good argument. Well I have no proof of it. – serge Jul 20 '16 at 19:13
  • @carlosarturoHurtado But $q=11\not\equiv 1\mod p=29$? – serge Jul 20 '16 at 19:16
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    @serge i have a mistake is $q\not\equiv 1\mod p$. sorry a lot. – carlos arturo Hurtado Jul 20 '16 at 19:28
  • which Algebra book you mean? – serge Jul 20 '16 at 19:30
  • and do you mean $p>q, p \not\equiv 1\mod q$? – serge Jul 20 '16 at 19:34
  • yes, sorry about that – carlos arturo Hurtado Jul 20 '16 at 20:07

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