A note on fixed points of automorphisms of infinite groups

Document Type : Research Paper

Authors

1 University of Napoli Federico II

2 National University of Ireland

3 Seconda Universita di Napoli

Abstract

‎Motivated by a celebrated theorem of Schur‎, ‎we show that if $\Gamma$ is a normal subgroup of the full automorphism group $Aut(G)$ of a group $G$ such that $Inn(G)$ is contained in $\Gamma$ and $Aut(G)/\Gamma$ has no uncountable abelian subgroups of prime exponent‎, ‎then $[G,\Gamma]$ is finite‎, ‎provided that the subgroup consisting of all elements of $G$ fixed by $\Gamma$ has finite index‎. ‎Some applications of this result are also given.‎

Keywords

Main Subjects


R. Baer (1952). Endlichkeitskriterien f\"ur Kommutatorgruppen. Math. Ann.. 124, 161-177 H. Dietrich and P. Moravec (2011). On the autocommutator subgroup and absolute centre of a group. J. Algebra. 341, 150-157 S. Franciosi and F. de Giovanni (1986). A note on groups with countable automorphism groups. Arch. Math. (Basel). 47, 12-16 P. Hegarty (1994). The absolute centre of a group. J. Algebra. 169, 929-935 D. J. S. Robinson (1972). Finiteness Conditions and Generalized Soluble Groups. Springer, Berlin. I. Schur (1904). \"Uber die Darstellung der endlichen Gruppen durch gebrochene linear Substitutionen. J. Reine Angew. Math.. 127, 20-50 R. F. Turner Smith (1964). Marginal subgroup properties for outer commutator words. Proc. London Math. Soc. (3). 14, 321-341 D. J. S. Robinson (1979). Infinite torsion groups as automorphism groups. Quart. J. Math. Oxford Ser. (2). 30, 351-364