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.‎

Keywords:

automorphism group , Schur's theorem , absolute centre