for $N > 0$, I'm trying to show Fermat's little theorem, for $3$ using the orbit stabilizer theorem:
$N^3 - N$ an element of $3\mathbb{Z}\ (3 \mod \mathbb{Z})$
Pf/ we can break it down into multiple cases
case 1: diagonal $i = j = k$ $(i,i,i)$ and orbit of this is $1 \times N$
case 2: $i$ not equal to $j$, but $j = k$
case 3: $i$ not equal to $j$ or $k$ and $j$ not equal to $k$
Can someone help me organize this and explain what's going on?
Thanks