On groups all of whose Haar graphs are Cayley graphs

TL;DR

This paper characterizes finite inner abelian groups for which all Haar graphs are Cayley graphs, showing only specific dihedral and quaternion groups qualify, and demonstrates that non-solvable groups have Haar graphs that are not Cayley graphs.

Contribution

It identifies the exact finite inner abelian groups with the property that all their Haar graphs are Cayley graphs, and highlights a distinction for non-solvable groups.

Findings

D6, D8, D10, and Q8 are the only such groups.

Non-solvable groups have Haar graphs that are not Cayley graphs.

Provides a characterization linking Haar graphs and Cayley graphs for specific groups.

Abstract

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such that ${\rm Aut}(\Sigma)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(\Sigma)$ and the $H$-orbits are equal to the bipartite sets of $\Sigma$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.