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 ℤ[G/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.
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Supported by RFBR (project No. 09-01-00099), by the Russian Ministry of Education through the Analytical Departmental Target Program “Development of Scientific Potential of the Higher School of Learning” (project No. 2.1.1/419), and by the Federal Program “Scientific and Scientific-Pedagogical Cadres of Innovative Russia” in 2009–2013 (gov. contract No. 02.740.11.5191).
Translated from Algebra i Logika, Vol. 49, No. 6, pp. 803–818, November-December, 2010.
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Romanovskii, N.S. Coproducts of rigid groups. Algebra Logic 49, 539–550 (2011). https://doi.org/10.1007/s10469-011-9116-y
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DOI: https://doi.org/10.1007/s10469-011-9116-y