This work is licensed under Creative Commons Attribution–NonCommercial International License
(CC BY-NC 4.0).
We prove that the group of automorphisms Aut(B(m,n)) of the free Burnside group B(m,n) is complete for every odd exponent n≥1003 and for any m>1, that is it has a trivial center and any automorphism of Aut(B(m,n)) is inner. Thus, the automorphism tower problem for groups B(m,n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, we obtain that the group of all inner automorphisms Inn(B(m,n)) is the unique normal subgroup in Aut(B(m,n)) among all its subgroups, which are isomorphic to free Burnside group B(s,n) of some rank s.