I wish to ask a problem in periodic strings, it might be well-known but I am a beginner in this subject, so I am very glad if someones can show me. My problem is that can we add some string to the end of a periodic string ($S$) to have a new periodic string ($S'$)? If yes, what are the conditions?

More formally, given $$S = [a_1 a_2 ... a_n]_t$$ - S is a periodic string (a string of $t$ repeated string $a_1 a_2 ... a_n$), where $n > 1$ and $t > 1$ and $a_i∈ A$ where $A$ is the alphabet (here we only consider the minimal $n$ such that $S$ is a period string - there does not exist $n'<n$ such that $S$ is a period string of consecutive strings of length $n'$). Suppose that if we concatenate a string [$c_1 c_2...c_k$] to $S$ ($c_j∈ A$ for all $j = 1,...,k$), then we have: $$S' = [a_1 a_2 ... a_n]_t [c_1 c_2...c_k].$$ So, does there exist the positive integers $m, s (m>1,s>1)$ such that $S' = [a_{\pi(1)}...a_{π(m)}]_s$ - another periodic string, where $π$ is some permutation from $\mathbf N$ to $\mathbf N$? If yes, what are the conditions? And are there some good algorithms to test this problem?

We see a trivial case when $[c_1 c_2...c_k] = [a_1 a_2 ... a_n]_v$ for any $v \in \mathbf N^*$. Do we have other cases? Could someone show me some researchs in this problem?

Thank you very much for your attention!