Prove that every non-zero element of $\Bbb Z_p$ has an order which divides $p-1$.

Question

How do I prove this if I am given the information that: $p \ge 2$ and is a prime and $g$ is an element of order $p − 1$ in $\Bbb Z_p$.

I know I am supposed to use Fermat's little theorem and do a contradiction proof but I am not quite sure on how to do it.

score 0 · Answer 1 · answered Sep 26 '20 at 14:30

0

Hint: if, by Fermat's little theorem, $a^p = a$ in $\mathbb{Z}_p$... what can you say about $a^{p-1}$?

answered Sep 26 '20 at 14:30

I thought $a^p-1 = 1 (mod p)$ was fermat's little theorem, but going by what you wrote $a^p-1$ does not equal $a$ i guess? – Jopps Sep 26 '20 at 14:54
They are equivalent statements. If you use the one you wrote, isn't the proof trivial by the definition of "order of an element"? – AnalysisStudent0414 Sep 26 '20 at 15:05
@AnalysisStudent0414 What definition of "order" are you using that makes it trivial? – Bill Dubuque Sep 26 '20 at 17:39

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