On automorphisms of Haar graphs of abelian groups

Authors

DOI:

https://doi.org/10.26493/2590-9770.1391.f46

Keywords:

automorphism group, Haar graph, s-arc

Abstract

Let G be a group and S ⊆ G. In this paper, a Haar graph of G with connection set S has vertex set ℤ2 × G and edge set {(0,g)(1,gs) : g ∈ G and s ∈ S}. Haar graphs are then natural bipartite analogues of Cayley digraphs, and are also called BiCayley graphs. We first examine the relationship between the automorphism group of the Cayley digraph of G with connection set S and the Haar graph of G with connection set S. We establish that the automorphism group of a Haar graph contains a natural subgroup isomorphic to the automorphism group of the corresponding Cayley digraph. In the case where G is abelian, we show there are exactly four situations in which the automorphism group of the Haar graph can be larger than the natural subgroup corresponding to the automorphism group of the Cayley digraph together with a specific involution, and analyze the full automorphism group in each of these cases. As an application, we show that all s-transitive Cayley graphs of generalized dihedral groups have a quasiprimitive automorphism group, can be constructed from digraphs of smaller order, or are Haar graphs of abelian groups whose automorphism groups have a particular permutation group theoretic property.

Published

2022-07-18

Issue

Section

The Marston Conder Issue of ADAM