Macedońska, Olga; Tomaszewski, Witold Centralizers of automorphisms permuting free generators. (English) Zbl 1435.20032 Open Math. 17, 15-22 (2019). Summary: By \(\sigma \in S_{km }\) we denote a permutation of the cycle-type \(k^m\) and also the induced automorphism permuting subscripts of free generators in the free group \(F_{km}\). It is known that the centralizer of the permutation \(\sigma\) in \(S_{km}\) is isomorphic to a wreath product \(Z_k \wr S_m\) and is generated by its two subgroups: the first one is isomorphic to \(Z_k^m\), the direct product of \(m\) cyclic groups of order \(k\), and the second one is \(S_m\). We show that the centralizer of the automorphism \(\sigma \in \operatorname{Aut}(F_{km})\) is generated by its subgroups isomorphic to \(Z_k^m \) and \(\operatorname{Aut}(F_m)\). MSC: 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups 20F28 Automorphism groups of groups Keywords:permutation; centralizer; automorphism PDFBibTeX XMLCite \textit{O. Macedońska} and \textit{W. Tomaszewski}, Open Math. 17, 15--22 (2019; Zbl 1435.20032) Full Text: DOI References: [1] Vogtmann K., Automorphisms of free groups and outer space, Geometriae Dedicata, 2002, 94, 1-31. · Zbl 1017.20035 [2] Boutin D.L., When are centralizers of finite subgroups of Out(F_n) finite? In: Groups Languages and Geometry (South Hadley, MA, 1998), Contemp. Math. 250, Amer. Math. Soc., Providence, RI, 1999, 37-58. · Zbl 0953.20033 [3] Pettet M., Virtually free groups with finitely many outer automorphisms, Trans. Amer. Math. Soc., 1997, 349 no.11, 4565-4587. · Zbl 0891.20026 [4] Algom-Kfir Y., Pfaff C., Normalizers and centralizers of cyclic subgroups generated by lone axis fully irreducible outer automorphisms, New York Journal of Mathematics, 2017, Vol. 23, 365-381. · Zbl 1386.20019 [5] Krstič S., Finitely generated virtually free groups have finitely presented automorphism group, Proc. London Math Soc. (3), 1992, 64(1), 49-69. · Zbl 0773.20008 [6] Kroshko N.V., Sushchansky V.I., Direct limits of symmetric and alternating groups with strictly diagonal embeddings, Archiv der Math., 1998, 71, 173-182. · Zbl 0929.20031 [7] Lipscomb S.L., The structure of the centralizer of a permutation, Semigroup Forum, 1988, 37 (3), 301-312. · Zbl 0641.20041 [8] Magnus W., Karrass A., Solitar D., Combinatorial group theory, 1966, New York-London-Sydney: Interscience Publishers. [9] Cohen R., Classes of automorphisms of free groups of infinite rank, Trans. Amer. Math. Soc., 1973, 177, 99-120. · Zbl 0278.20033 [10] Neumann B., Die Automorphismengruppe der freien Gruppen, Math Ann., 1932, 107, 367-386. · JFM 58.0124.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.