Partitions of groups into large subsets

2014/08/24 by Protasov, Igor, Slobodianiuk, Sergii · 1 citation
#03E05 #05A18 #20B07 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1408.5606

Abstract

Let G be a group and let k be a cardinal. A subset A of G is called left (right) k-large if there exists a subset F of G such that |F| < and G = FA (G = AF). We say that A is k-large if A is left and right k-large. It is known that every infinite group G can be partitioned into countably many ℵ₀-large subsets. On the other hand, every amenable (in particular Abelian) group G cannot be partitioned into > ℵ₀ ℵ₀-large subsets. We prove that every infinite group G of cardinality k can be partitioned into k left- ℵ₁-large subsets and every free group F_k in the infinite alphabet k can be partitioned into k 4-large subsets.

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