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January 15th, 2018, 02:11 AM   #1
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Order of an element in a Group

Q : An element 'a' in a group has order 100. Then what is the order of a^65 ?

Property : if o(a) = n & p is prime to n then o(a^p) = n.
Using the above property, removing common factors between 65 & 100, we get 13 & 20.

Then o(a^65) would be 20 ?

I just tried let me know how to solve above question if it's wrong.
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January 17th, 2018, 02:44 AM   #2
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Good morning.

Yes, your answer is correct. It is a result of the following property, which generalizes the one you mentioned in your post:

Let $\displaystyle o(a)=n$ and let $\displaystyle m\in{\mathbb N}$.
Then $\displaystyle o\left(a^m\right)=\dfrac nd$, where $\displaystyle d=\gcd(n,m)$.

Last edited by skipjack; January 17th, 2018 at 04:52 PM.
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January 17th, 2018, 02:57 AM   #3
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Thank you so much for the response.

Last edited by skipjack; January 17th, 2018 at 04:53 PM.
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January 17th, 2018, 09:09 AM   #4
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Ould Yubba has given you the correct formula. The proof is straight forward:

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