Groups of outer finitary automorphisms. (Q650470)

An automorphism \(\alpha\) of a group \(G\) is called finitary if \(|G:C_G(\alpha)|<\infty\). It can be easily seen that the set \(\mathrm{FAut}(G)\) of all finitary automorphisms of a group \(G\) forms a subgroup of \(\Aut(G)\). The quotient group \(\mathrm{FAut}(G)/(\mathrm{FAut}(G)\cap\mathrm{Inn}(G))\) is called the group of outer finitary automorphisms of \(G\) and it is denoted by \(\mathrm{FOut}(G)\). It is proved that given any group \(G\), \(\mathrm{FOut}(G)\) is always locally finite.