Note: A conjecture on partitions of groups

Abstract

We conjecture that every infinite group G can be partitioned into countably many cells G = n∈ω A n such that cov(A n A −1 n ) = |G| for each n ∈ ω. Here cov(A) = min{|X| : X ⊆ G, G = XA}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.2010 Mathematics Subject Classification. 03E05, 20B07, 20F69.

Partitions of groups into large subsets

“…Every infinite group G of regular cardinality κ can be partitioned G = A 1 ∪ A 2 such that A 1 and A 2 are not left κ-large. In [10] we show that this statement fails to be true for every Abelian group of singular cardinality κ.…”

Functional Boundedness of Balleans: Coarse Versions of Compactness

“…Following [20], we define group G to be κ-normal if every subset F ∈ [G] <κ is contained in a normal subgroup H ∈ [G] <κ of G.…”

Partitions of Groups

“…For λ = ℵ 0 , Theorem 8.1 was proved in [22] to partition each infinite totally bounded topological group G into |G| dense subsets. The results of this section are from preprint [42]. More on partitions of topological groups into dense subsets see [25] and [14,Chapter 13].…”