, Volume 99, Issue 5, pp 425-432,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 16 Nov 2012

Fixed points of automorphisms preserving the length of words in free solvable groups

Abstract

Let δ be an automorphism of prime order p of the free group F n . Suppose δ has no fixed points and preserves the length of words. By σ := δ (m) we denote the automorphism of the free solvable group \({F_{n}/F_n^{(m)} }\) induced by δ. We show that every fixed point of σ has the form \({cc^{\sigma} \ldots c^{\sigma^{p-1}}}\) , where \({c\in F_n^{(m-1)}/F_n^{(m)}}\) . This is a generalization of some known results, including the Macedońska–Solitar Theorem [10].