Outer automorphisms of locally finite p-groups

Every group is an outer automorphism group of a locally finite p-group. This extends an earlier result [M. Dugas, R. Göbel, On locally finite p-groups and a problem of Philip Hallʹs, J. Algebra 159 (1) (1993) 115–138] about countable outer automorphism groups. It is also in sharp contrast to results concerning the existence of outer automorphisms of nilpotent groups in [W. Gaschütz, Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen, J. Algebra 4 (1966) 1–2; O. Puglisi, A note on the automorphism group of a locally finite p-group, Bull. London Math. Soc. 24 (5) (1992) 437–441; A.E. Zalesskii, A nilpotent p-group has an outer automorphism, Dokl. Akad. Nauk SSSR 196 (1971) 751–754].