A note on fixed points of automorphisms of infinite groups

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.