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 $$\mathfrak{G} = \mathfrak{N}\mathfrak{A}$$ 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 $$\mathfrak{G}$$ .