THE AUTOMORPHISM TOWER PROBLEM FOR FREE PERIODIC GROUPS

Abstract

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 $n\geq 1003$ 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 $s$.