Coproducts of rigid groups

Let ε = (ε ₁, . . . , ε _m ) be a tuple consisting of zeros and ones. Suppose that a group G has a normal series of the form G = G ₁ ≥ G ₂ ≥ . . . ≥ G _m ≥ G _m+1 = 1, in which G _i > G _i+1 for ε _i = 1, G _i = G _i+1 for ε _i = 0, and all factors G _i /G _i+1 of the series are Abelian and are torsion free as right ℤG/G _i ]-modules. Such a series, if it exists, is defined by the group G and by the tuple ε uniquely. We call G with the specified series a rigid m-graded group with grading ε. In a free solvable group of derived length m, the above-formulated condition is satisfied by a series of derived subgroups. We define the concept of a morphism of rigid m-graded groups. It is proved that the category of rigid m-graded groups contains coproducts, and we show how to construct a coproduct G◦H of two given rigid m-graded groups. Also it is stated that if G is a rigid m-graded group with grading (1, 1, . . . , 1), and F is a free solvable group of derived length m with basis {x ₁, . . . , x _n }, then G◦F is the coordinate group of an affine space G ⁿ in variables x ₁, . . . , x _n and this space is irreducible in the Zariski topology.