Splitting via Noncommutativity

TL;DR

This paper introduces the concept of strict n-split decompositions in nonabelian groups, classifies groups with such decompositions for small n, and explores bounds on subgroup indices.

Contribution

It defines strict n-split decompositions, classifies finite groups with these decompositions for n=1,2,3, and establishes bounds on subgroup indices based on n.

Findings

Every finite nonabelian group admits a strict n-split decomposition for some n.

Complete classification of groups with strict n-split decompositions for n=1, 2, 3.

Bound on the index of the abelian subgroup in terms of n.

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 $B_1, B_2, \ldots, B_n$, 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 $n$.