Hypothesis: $\sum\limits_{n \in A} ccl_p(n) = p-1 $

Question

Let G be a the finite set of integers $1, \dots , p-1 $

One can see, if $|G| = p-1 $ with $p$ prime, then $\forall a \in G: \exists n \in \mathbb{N}$ with $ a^n = 1$ mod $p$ and $ 1 \le n < p $

I define: $cl_p (a) := n $, I played a bit around with my definition and found something, which made me curious:

I claim:

$A := \{ n \in \mathbb{N} | a^n = 1, 1 \le a <p \} $

$ccl_p(n) : = |\{x \in \mathbb{N} | 1 \le x < p \text{ with } x^n = 1 \} |$, $n \in A$

$\sum\limits_{n \in A} ccl_p(n) = p-1 $

My mathematical capabilities are rather low, so I could not construct a prove or a counterexample, however I wrote a first java program to test my hypothesis and could not a case, when my hypothesis was wrong. So I am actually quite stunned.(how to attach a txt file (?))

Now I would really like / appreciate any sort of constructive help / assistance in either proving or disproving my hypothesis. All assistance is appreciated. (or recommending a better title)

Code here: http://hostcode.sourceforge.net/view/4851

Answer 1

One can see, if $|G| = p $ with $p$ prime, then $\forall a \in G: \exists n \in \mathbb{N}$ with $ a^n = 1$ and $ 1 \le n < p $

I certainly cannot see that. In fact, that statement is clearly false for all prime values of $p$, because if $|G|=p$ and $p$ is prime, then $(G,\cdot)$ is isomorphic to $(\mathbb Z_p, +)$ and your statement is equal to saying that for all $x\in\mathbb Z_p$, there exists some $n\in\mathbb N$ such that $n\cdot x=0$ and $1\le n < p$ which is not true. In fact, if $n<p$, then $n\cdot x=0$ is true if and only if $x=0$.