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 $3$.

Keywords

• n-Abelian partition,commuting graph

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