GROUP PARTITIONS VIA COMMUTATIVITY

Abstract

Let $G$ be a nonabelian group, $A\subset G$ an abelian subgroup and $n\geqslant 2$ an integer. We say that $G$ has an $n$-abelian partition with respect to $A$, if there exists a partition of $G$ into $A$ and $n$ disjoint commuting subsets $A_1, A_2,\ldots, A_n$ of $G$, such that $|A_i|>1$ for each $i=1, 2, \ldots, n$. We classify all nonabelian groups, up to isomorphism, which have an $n$-abelian partition, for $n=2$ and $3$.