The embedding problem with metabelian kernel

The relationship between the solvability conditions for the embedding problem with metabelian kernel over a p-extension of fields and the associated embedding problem of the first kind with a maximal p-group is studied. It is proved that these problems are equivalent. Bibliography: 1title.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 97–99.The present paper was supported by the Russian Foundation for Basic Research, grant No. 96-01-00854.     

Localizability of the embedding problem with symplectic kernel

In this paper we consider the question of how much information is supplied by local solutions to a global embedding problem for the special case in which the normal subgroup belonging to the given group extension is the projective symplectic group PSp(2m, q). It is proved that for suitable Galois extensions K of a given number field k (which constitute part of the data of the embedding problem), the local solutions in a sense determine whether or not an extension K ? K, Galois over k, with $\text{G(}\text{L}\text{K}\text{) ≈ PSp(2m, q)}$, represents a global solution to the embedding problem.     

On the embedding problem with non-Abelian kernel of orderp 4. V

A survey of solvability conditions for the embedding problem of number fields, in which the kernel is a non-Abelian group of order p⁴, is completed. As a kernel, the two 2-groups with two generators a, b and with the following relations are considered: a ⁸ =1, b ² =1, [a,b]=a⁻² in the first group, and a ⁸ =1, b ² =a ⁴ , [a,b]=a⁻² in the second. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 127–132. Translated by V. V. Ishkhanov.     

On the embedding problem with non-Abelian kernel of orderp 4. IV

In the present paper, the embedding problem is considered for number fields with p-groups whose kernel is either of two groups with two generators α and β and with the following relations: (1) α^ρ=1, α^ρ=1, [α,β,β]=1, [α,β,α,α]=1, or (2) α^ρ=[α, β α], β^ρ=1, [α,β,β]=1. It is shown that for the solvability of the original embedding problem it is necessary and sufficient to have the solvability of the associated Abelian and local problems for all completions of the base fields. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 120–126. Translated by V. V. Ishkhanov.     

A case of imbedding problem with a solvable kernel

Journal of Mathematical Sciences - It is shown for the solution of the imbedding problem of algebraic number fields with a finite solvable kernel one can take a field if the orders of the kernel...     

On the embedding problem with noncommutative kernel of order p4. VI

In the case of number fields the embedding problem of a p-extension with non-Abelian kernel of order p⁴ is studied. The two kernels of order 3⁴ with generators α, γ and relations α⁹ = 1, [α,α]³=1,[α,αγγ]==1,[αγγ]=α³,γ³=1 or γ³=α³ and the kernel of order 2⁴ with generators α, β, γ and relations α⁴=1 β²,[αγ]=1, [α,γ]=1,[βγ]=α² are considered. For kernels of odd order the embedding problem is always solvable. For the kernel of order 16 the solvability conditions are reduced to those for the associated problems at the Archimedean points, and to the compatibility condition. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 74–82.     

The embedding problem with non-Abelian kernel for local fields

The embedding problem of local fields with p-groups is equivalent to its associated Abelian problem if the inequality d ≥ r + 2 is valid; here d and r are the numbers of generators of the Demushkin group and of the Galois group of an embedded field. Bibliography: 6 titles.     

The embedding problem with kernel PSL (2, p 2)

The embedding problem of number fields is considered. It is proved that the problem is solvable if and only if all associated local problems corresponding to infinite points are solvable. It is also proved that the solvability of the adjoined problem with Sylow 2-group implies the solvability of the original problem. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 135–145.     

On the problem of field embedding

A cohomological interpretation of solvability conditions for the problem of field embedding is given. Embedding conditions are formulated in these terms in cases when there are additional restrictions on the constructed field (e.g., fixation of the branching points of the field in the number case).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 137–140, August, 1968.     

On a straight-line embedding problem of graphs

Let G be a planar graph with n vertices, v be a specified vertex of G, and P be a set of n points in the Euclidian plane in general position. A straight-line embedding of G onto P is an embedding of G onto whose images of vertices are distinct points in P and whose images of edges are (straight) line segments. In this paper, we classify into five classes those pairs of G and v such that for any P and any p P, G has a straight-line embedding onto P which maps v to p. We then show that all graphs in three of the classes indeed have such an embedding. This result gives a solution to a generalized version of the rooted-tree embedding problem raised by M. Perles.     

Completely solvable imbedding problems with Abelian kernel for local fields

The imbedding problem for p-groups is considered in which the fields are assumed to be local and the kernel commutative. Additional conditions are investigated under which a solvable imbedding problem has a field as solution. Sufficient conditions are found for such solvability in the form of inequalities imposed on the number of generators of certain groups.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 67–73, 1978.     

On the residual finiteness of free products of solvable minimax groups with cyclic amalgamated subgroups

A necessary and sufficient condition for the residual finiteness of a (generalized) free product of two residually finite solvable-by-finite minimax groups with cyclic amalgamated subgroups is obtained. This generalizes the well-known Dyer theorem claiming that every free product of two polycyclic-by-finite groups with cyclic amalgamated subgroups is a residually finite group.     

On embedding infinite cyclic covers in compact 3-manifolds

We show that there are knots whose infinite cyclic covers do not embed in any compact 3-manifold, answering a question of Jiang et al. (2006).