# A group $G$ of order $n$ is cyclic iff $G$ contains an element of order $n$ [closed]

Let $G$ be a finite group of order $n$. Prove that $G$ is cyclic if and only if $G$ contains an element of order $n$.

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## closed as off-topic by Jonas Meyer, Krish, අරුණ, John, Najib IdrissiMar 26 at 8:01

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Which part is giving you trouble, "if" or "only if"? –  Robert Israel Sep 13 '11 at 18:47
This question does not show any research effort –  The Chaz 2.0 Sep 13 '11 at 19:09
if and only if is obviously wrong. –  Dinesh Sep 13 '11 at 19:57
@Dinesh: Which part do you think is wrong? –  Robert Israel Sep 13 '11 at 20:11
@Dinesh, did you overlook $G$ is of order $n$? –  Soarer Sep 13 '11 at 20:29

The key component in the proof is that if $a^i=a^j$ for some element $a$ then $a^{i-j}=e$ and hence the order of $a$ divides $i-j$. To say more will be simply to write a complete solution of the question for you.

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