Centralizers of automorphisms permuting free generators

Abstract

Abstract By σ ∈ Skm we denote a permutation of the cycle-type km and also the induced automorphism permuting subscripts of free generators in the free group Fkm. It is known that the centralizer of the permutation σ in Skm is isomorphic to a wreath product Zk ≀ Sm and is generated by its two subgroups: the first one is isomorphic to $\begin{array}{} \displaystyle Z_k^m \end{array}$, the direct product of m cyclic groups of order k, and the second one is Sm. We show that the centralizer of the automorphism σ ∈ Aut(Fkm) is generated by its subgroups isomorphic to $\begin{array}{} \displaystyle Z_k^m \end{array}$ and Aut(Fm).