On the product of an abelian group and a nilpotent group
We study the structure of the product of an Abelian group and a nilpotent group. Conditions for the existence of a normal subgroup in one of the factors are given. These conditions generalize the known results on the product of two Abelian groups. The statements obtained are used to describe the structure of a product of an infinite cyclic subgroup and a periodic nilpotent subgroup.
In this paper we address the celebrated Albert problem for exceptional Jordan algebras (i.e. Albert algebras): Does every Albert division algebra contain a cubic cyclic subfield? We prove that for any Albert division algebra $A$ over a field $k$ of arbitrary characteristic, there is a suitable isotope that contains a cubic cyclic subfield. It follows from this that for any Albert division algebra $A$ over a field $k$, the structure group $\text{\bf Str}(A)$ always contains a subgroup of type $^3D_4$ defined over $k$.