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If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$?

Could anybody clear my this doubt?

Thanks for the help

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Hint: Let order of element $a$ is $n$ then $order (a^m) = \dfrac {n}{(m, n)}$, where $(m, n)$ is the gcd of $m$ and $n$.

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    $\begingroup$ That mean we can have $12 = \dfrac{n}{(12, n)} then possible value of n will be be the answer? $\endgroup$ – monalisa Apr 30 '13 at 18:49
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    $\begingroup$ No, @monalisa. You know that $$12=\frac{n}{\gcd(5,n)}.$$ Given that $5$ is a prime there aren't too many choices for $\gcd(5,n)$. $\endgroup$ – Jyrki Lahtonen Apr 30 '13 at 19:01
  • $\begingroup$ @JyrkiLahtonen Oh I misread. Thanks . $\endgroup$ – srijan Apr 30 '13 at 19:14

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