Let 𝐺 be a nonabelian group.
We say that 𝐺 has an abelian partition if there exists a partition of 𝐺 into commuting subsets of 𝐺 such that for each .
This paper investigates problems relating to groups with abelian partitions.
Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition.
We also find bounds for the minimum number of partitions for several families of groups which admit abelian partitions – with exact calculations in some cases.
Finally, we examine how the size of a partition with the minimum number of parts behaves with respect to the direct product.
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