A metabelian pro-p-group G is rigid if it has a normal series of the form G = G1 ≥ G2 ≥ G3 = 1 such that the factor group A = G/G2 is torsion-free Abelian and C = G2 is torsion-free as a ZpA-module. If G is a non-Abelian group, then the subgroup G2, as well as the given series, is uniquely defined by the properties mentioned. An Abelian pro-p-group is rigid if it is torsion-free, and as G2 we can take either the trivial subgroup or the entire group. We prove that all rigid 2-step solvable pro-p-groups are mutually universally equivalent. Rigid metabelian pro-p-groups can be treated as 2-graded groups with possible gradings (1, 1), (1, 0), and (0, 1). If a group is 2-step solvable, then its grading is (1, 1). For an Abelian group, there are two options: namely, grading (1, 0), if G2 = 1, and grading (0, 1) if G2 = G. A morphism between 2-graded rigid pro-p-groups is a homomorphism \( \varphi \) : G → H such that Gi \( \varphi \) ≤ Hi. It is shown that in the category of 2-graded rigid pro-p-groups, a coproduct operation exists, and we establish its properties.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. S. Romanovskii, “Divisible rigid groups,” Algebra Logika, 47, No. 6, 762-776 (2008).
N. S. Romanovskii, “Equational Noetherianness of rigid soluble groups,” Algebra Logika, 48, No. 2, 258-279 (2009).
N. S. Romanovskii, “Irreducible algebraic sets over divisible decomposed rigid groups,” Algebra Logika, 48, No. 6, 793-818 (2009).
N. S. Romanovskii, “Coproducts of rigid groups,” Algebra Logika, 49, No. 6, 803-818 (2010).
A. Myasnikov and N. Romanovskiy, “Krull dimension of solvable groups,” J. Alg., 324, No. 10, 2814-2831 (2010).
A. G. Myasnikov and N. S. Romanovskii, “Universal theories for rigid soluble groups,” Algebra Logika, 50, No. 6, 802-821 (2011).
N. S. Romanovskiy, “Presentations for rigid solvable groups,” J. Group Th., 15, No. 6, 793-810 (2012).
S. G. Melesheva, “Equations and algebraic geometry over profinite groups,” Algebra Logika, 49, No. 5, 654-669 (2010).
G. Baumslag, A. Myasnikov, and V. Remeslennikov, “Algebraic geometry over groups. I: Algebraic sets and ideal theory,” J. Alg., 219, No. 1, 16-79 (1999).
A. Myasnikov and V. N. Remeslennikov, “Algebraic geometry over groups. II: Logical foundations,” J. Alg., 234, No. 1, 225-276 (2000).
J. S. Wilson, Profinite Groups, London Math. Soc. Mon., New Ser., 19, Clarendon, Oxford (1998).
O. Chapuis, “∀-free metabelian groups,” J. Symb. Log., 62, No. 1, 159-174 (1997).
V. N. Remeslennikov, “Embedding theorems for profinite groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 43, No. 2, 399-417 (1979).
N. S. Romanovskii, “Shmel’kin embeddings for abstract and profinite groups,” Algebra Logika, 38, No. 5, 598-612 (1999).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated from Algebra i Logika, Vol. 53, No. 2, pp. 162-177, March-April, 2014.
Rights and permissions
About this article
Cite this article
Afanas’eva, S.G., Romanovskii, N.S. Rigid Metabelian Pro-p-Groups. Algebra Logic 53, 102–113 (2014). https://doi.org/10.1007/s10469-014-9274-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-014-9274-9