Splitting via Noncommutativity

Abstract

Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1,B_2,\dots ,B_n$, such that $|B_i|>1$ for each $i$ and within each set $B_i$, no two distinct elements commute. We show that every finite nonabelian group has a strict $n$-split decomposition for some $n$. We classify all finite groups $G$, up to isomorphism, which have a strict $n$-split decomposition for $n=1,2,3$. Finally, we show that for a nonabelian group $G$ having a strict $n$-split decomposition, the index $|G:A|$ is bounded by some function of $n$.

31 pages