Group Partitions via Commutativity and Related Topics

TL;DR

This paper classifies nonabelian groups with specific partitions into abelian and commuting subsets, and explores the enumeration of spanning trees in their commuting graphs.

Contribution

It provides a classification of nonabelian groups with 2- and 3-abelian partitions and investigates spanning tree counts in their commuting graphs.

Findings

Classified all nonabelian groups with 2- and 3-abelian partitions.

Derived formulas for the number of spanning trees in commuting graphs.

Suggested methods for computing spanning trees in specific group cases.

Abstract

Let $G$ be a nonabelian group, $A\subseteq G$ 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 first classify all nonabelian groups, up to isomorphism, which have an $n$-abelian partition for $n=2, 3$. Then, we provide some formulas concerning the number of spanning trees of commuting graphs associated with certain finite groups. Finally, we point out some ways to finding the number of spanning trees of the commuting graphs of some specific groups.