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.‎

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