Some kinds of primitive and non-primitive words

Abstract

If the length of a primitive word $p$ is equal to the length of another primitive word $q$, then $p^{n}q^{m}$ is a primitive word for any $n,m\ge 1$ and $(n,m)\ne (1,1)$. This was obtained separately by Tetsuo Moriya in 2008 and Shyr and Yu in 1994. In this paper, we prove that if the length of $p$ is divisible by the length of $q$ and the length of $p$ is less than or equal to $m$ times the length of $q$, then $p^{n}q^{m}$ is a primitive word for any $n,m\ge 1$ and $(n,m)\ne (1,1)$. Then we show that if $uv,u$ are non-primitive words and the length of $u$ is divisible by the length $v$ or one of the length of $u$ and $uv$ is odd for any two nonempty words $u$ and $v$, then $u$ is a power of $v$.