In an earlier post, I asked for some intuition on Cauchy's Theorem for Groups (for every prime divisor of the order of a finite group, there exists an elements who's order is that prime). I got great answers and was made aware of this incredible proof by McKay. Personally I think its a gorgeous proof, and love the insight he provides in the opening paragraph. However, I'm having difficulty with a single claim he makes.
"Otherwise, if two components of a p-tuple are distinct, there are p members in the equivalence class."
To summarize his claim. suppose $p$ is a prime that divides the order of some group $G$ and consider the string
$$x_1x_2\dots x_p = e$$
then if there exists $i,j$ such that $x_i = x_j, i\ne j$ then for any cyclic permutation of these elements, the strings of elements are not identical.
I can't quite see how this conclusion can be made on the distinctness of a single pair of components. Wouldn't this claim imply that if $g_1g_2\dots g_p = e, g_i\ne e$ than all the $g_i$'s are distinct?
Thanks for the help as always!