Abelian Group Of Size Research Articles

Indecomposable solutions of the Yang-Baxter equation with permutation group of sizes pq and p 2 q

In this paper we study the problem of the classification of indecomposable solutions of the Yang-Baxter equation. Using a scheme proposed by Bachiller, Cedó, and Jespers, and recent advances in the classification of braces we classify all indecomposable solutions with some particular permutation groups. We do this for all groups of size pq, all abelian groups of size p 2 q and all dihedral groups of size p 2 q.

Eulerian Subgraphs and S-connectivity of Graphs

Abstract Collapsible graphs are introduced by Caltin to study Eulerian subgraphs, and S-group-connectivity is introduced by Jaeger et al. to study flows of graphs. Lai established a connection of those graph classes by showing that collapsible graphs have S-connectivity for group S of order 4. In a survey paper in 2011, Lai et al. conjectured that this property holds for all finite Abelian groups of size at least 4. We prove this conjecture for all groups of even order |S| ≥ 4 and of large odd order |S| ≥ 53.

Splitting kernels into small summands

Let λ be a regular cardinal. An epimorphism between abelian groups is λ-pure if it is projective with respect to abelian groups of size less than λ. We show that cotorsion groups A have λ-pure projective dimension greater than 1 for all uncountable λ ≤ |A/tA|, where tA denotes the torsion subgroup of A. For λ > |A/tA|, cotorsion groups A are λ-pure projective.

The number of openly generated Boolean algebras

AbstractThis article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly σ-filtered Boolean algebras.We show that for every uncountable regular cardinal κ there are 2κ pairwise non-isomorphic openly generated Boolean algebras of size κ > ℵ1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolean algebras constructed here are almost free.Moreover, for every infinite regular cardinal κ we construct 2κ pairwise non-isomorphic Boolean algebras of size κ that are tightly σ-filtered and c.c.c.These two results contrast nicely with Koppelberg's theorem in [12] that for every uncountable regular cardinal κ there are only 2κ isomorphism types of projective Boolean algebras of size κ.