A note on finite embedding problems with nilpotent kernel

A note on some embedding problems for oriented graphs

We conjecture that every oriented graph G on n vertices with δ(G), δ−(G) ≥ 5n/12 contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing of transitive triangles in an oriented graph. A link between Ramsey numbers and perfect packings of transitive tournaments is also considered. 1.1. Powers of Hamilton cycles. One of th...

Embedding Problems and Finite Quotients

We give a condition on a family of solutions of quotients of an embedding problem which implies the embedding problem has a solution. This shows, in particular, that to solve an embedding problem associated to the maximal extension of a number field unramified outside a fixed finite set of places, it suffices to find a solution for each finite quotient of the embedding problem. This statement i...

A note on extensions of nilpotent groups

A further advance—though it was not presented as such—was contained in Theorem 1.6 of [1], which effectively gave an example showing that the answer to our question is not always affirmative. The main purpose of this paper is to generalize both Theorem 0.1 and the counterexample contained in Theorem 1.6 of [1]. Thus we prove (cf. Theorems 2.1 and 2.3, which reproduce part (b) and part (a) of Th...

A Note on Nilpotent Lattice Matrices

Some properties and characterizations for nilpotent matrices are established and in particular, a necessary and sufficient condition for an n × n nilpotent matrix to have the nilpotent index 2 and 3 is given. Mathematics Subject Classification: 15A33; 06D99

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup