Group bases for some solvable groups and semidirect products

TL;DR

This paper investigates the existence of bases in finite groups, showing that many semidirect products, including all with abelian factors, have bases, while the quaternions do not.

Contribution

It establishes conditions under which certain finite groups and semidirect products possess bases, expanding understanding of group bases beyond abelian cases.

Findings

Semidirect products of finite groups have bases.

All groups of order m or 2m with odd, cube-free m have bases.

The quaternions do not have a basis.

Abstract

A set $B$ is a basis for a vector space $V$ if every element of $V$ can be uniquely written as a linear combination of the elements of $B$. There is a similar definition of a basis for a finite group. We show that certain semidirect products of finite groups---including all semidirect products of finite abelian groups---have bases; any group of order $m$ or $2m$ for odd, cube-free $m$ has a basis; and the quaternions do not have a basis.