Some properties of periodic words (Q1810194)

It is shown that if \(T\) is a simple (i.e., primitive) word such that \(T^3=XY\) for some words \(X\) and \(Y\), then \(X\) or \(Y\) has the same minimum period as \(T\) does. The proof uses elementary combinatorial arguments on words, and it can be simplified by considering Lyndon words. Indeed, \(T^3=Z_1L^2Z_2\), where \(L\) is a Lyndon word conjugate to \(T\). Hence \(T^3=XY\) implies that either \(X\) or \(Y\) contains \(L\) as a factor. The claim follows from this, since the minimum period of \(L\) is its length. A similar result is proven for the case \(T^2=XY\) (with \(T\) simple).