The directed Cayley diameter and the Davenport constant

TL;DR

This paper introduces the concept of directed geodesic atoms to study the directed Cayley diameter of finite groups, providing bounds, algorithms, and computations for various groups, and relates it to Davenport constants.

Contribution

It defines directed geodesic atoms, develops algorithms for their computation, and extends the understanding of Cayley diameters and Davenport constants for non-abelian groups up to order 42.

Findings

Bounds for directed Cayley diameter established.

Algorithm implemented in GAP for group analysis.

Extended Davenport constant computations for multiple groups.

Abstract

The directed Cayley diameter of a finite group is investigated in terms of the monoid of product-one sequences over the group, via the new notion of directed geodesic atoms. Two quantities associated to the set of directed geodesic atoms provide lower and upper bounds for the directed Cayley diameter. An algorithm for computing the directed geodesic atoms is implemented in GAP, and is applied to determine the above mentioned quantities for all non-abelian groups of order at most $42$, and for the alternating group of degree $5$. Furthermore, the small and large Davenport constants of all these groups are computed (excepting the large Davenport constant for $A_5$), extending thereby the formerly obtained results on the groups of order less than $32$. Along the way the directed Cayley diameter of a finite abelian group is expressed in terms of its invariants.