Partitions of groups into sparse subsets

Igor Protasov

Algebra and discrete mathematics, Tome 13 (2012) no. 1, pp. 107-110.

Résumé

A subset $A$ of a group $G$ is called sparse if, for every infinite subset $X$ of $G$, there exists a finite subset $F\subset X$, such that $\bigcap _{x\in F}xA$ is finite. We denote by $\eta (G)$ the minimal cardinal such that $G$ can be partitioned in $\eta (G)$ sparse subsets. If $|G|>({\kappa }^{+}{)}^{{\mathrm{\aleph }}_0}$ then $\eta (G)>\kappa$, if $|G|⩽{\kappa }^{+}$ then $\eta (G)⩽\kappa$. We show also that $cov(A)⩾cf|G|$ for each sparse subset $A$ of an infinite group $G$, where $cov(A)=min\{|X|:G=XA\}$.

Détail

Keywords: sparse subset of a group.
Mots-clés : partition of a group

Bibliographie

[1] T. Banakh, I. Protasov, “Partition of groups and matroids into independent subsets”, Algebra and Discrete Math., 10 (2010), 1–7 | MR | Zbl

[2] M. Filali, Ie. Lutsenko, I. V. Protasov, “Boolean group ideals and the ideal structure of $\beta G$”, Math. Stud., 31 (2009), 19–28 | MR | Zbl

[3] Ie. Lutsenko, I. V. Protasov, “Sparse, thin and other subsets of groups”, Intern. J. Algebra Computation, 19 (2009), 491–510 | DOI | MR | Zbl

[4] I. V. Protasov, “Partition of groups into large subsets”, Math. Notes, 73 (2003), 271–281 | DOI | MR

[5] I. V. Protasov, “Small systems of generators of groups”, Math. Notes, 76 (2004), 420–426 | DOI | MR | Zbl

[6] I. Protasov, “Partition of groups into thin subsets”, Algebra and Discrete Math., 11 (2011), 88–92 | MR

[7] I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., 12, VNTL, Lviv, 2007 | MR | Zbl