It is proved that if a periodic group $\mathfrak G$ has an extremal normal divisor $\mathfrak N$ , determining a complete abelian factor group $\mathfrak G/\mathfrak N$ , then the center of the group $\mathfrak G$ contains a complete abelian subgroup $\mathfrak A$, satisfying the relation $\mathfrak G=\mathfrak{NA}$ and intersecting $\mathfrak N$ on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group $\mathfrak G$ is a finite extension of a contained in it subgroup of inner automorphisms of the group $\mathfrak G$.