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October 25th, 2017, 11:12 AM   #1
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Order of the residue classes

There is one problem I can't solve for the particular section under prime power modulus in my textbook, and it's this:

If the reduced residue classes $a$ and $b \,($mod $p$) both have order $3^j,$ how can I show that the two residue classes $ab$ and $ab^2$ one of them has order $3^j$ and the other $3^{j^{'}},$ where $p$ is prime and $j>0$, for $j^{'}<j$ ?

How could I show this? Could anyone help me please?

Last edited by skipjack; October 25th, 2017 at 11:50 AM.
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October 26th, 2017, 05:49 PM   #2
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Your problem is really one about finite cyclic groups. For a prime p, the integers modulo a power of p have a primitive root; i.e. the underlying group is cyclic. Here's your problem restated:

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November 4th, 2017, 07:11 AM   #3
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Thank you very much !
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