If a string $S$ can't itself be broken down into into repeating strings, then how can we prove that the no. of friends of $S$ is equals to its length?

Question

Truly speaking the rule we have to prove is below which is taken from the Wikipedia page which doesn't gives its proof.

If $S$[a given string] is built up of several copies of the string $T$, and $T$ cannot itself be broken down further into repeating strings, then the number of friends of $S$ (including $S$ itself) is equal to the length of $T$.

Here, friend of a string $S$ is the set of strings which are identical to $S$ when it's end point are joined.

Wikipedia used this fact to prove Fermat's little theorem but didn't gave any proof of it.

Answer 1

Consider the friends of $T$. Since $T$ is not composed of repeating strings, each unit shift of $T$ will be distinct, so $T$ has as many friends as the the length ($:=L$) of $T$. Use a subscript on $T$ to denote the number of unit shifts for each friend of $T$, noting $T=T_0=T_L$.

Now the friends of $S$ will be composed of copies of these friends of $T$. As such there are only $L$ distinct friends of $S$, since $S_L=S$.