Generalized dihedral CI-groups

Ted Dobson,Pablo Spiga,Mikhail Muzychuk

doi:10.26493/1855-3974.2443.02e

Abstract

In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if R is a generalised dihedral group and if R is a CI-group, then for every odd prime p the Sylow p-subgroup of R has order p, or 9. Consequently, any CI-group with quotient a generalised dihedral group has the same restriction, that for every odd prime p the Sylow p-subgroup of the group has order p, or 9.

Highlights

Let R be a finite group and let S be a subset of R
If R is a generalised dihedral group and if R is a CI-group, for every odd prime p the Sylow p-subgroup of R has order p, or 9
The group R is a DCI-group if Cay(R, S) is a DCI-graph for every subset S of R

Summary

Introduction

Let R be a finite group and let S be a subset of R. Let Dih(A) be a generalised dihedral group over the abelian group A. If Dih(A) is a CI-group, for every odd prime p the Sylow p-subgroup of A has order p, or 9. If Dih(A) is a DCI-group, in addition, the Sylow 3-subgroup has order 3. There is a corresponding isomorphism problem for Haar graphs, and if the group A is abelian, it is equivalent to the isomorphism problem for Cayley graphs of generalised dihedral groups Dih(A) that are bipartite (for nonabelian groups the problems are not equivalent, as for non-abelian groups Haar graphs need not be transitive), see [17, Lemma 2.2]. If isomorphic bipartite Cayley graphs of Dih(A) are isomorphic by group automorphisms of A, we say A is a BCI-group. We will show that Zk3 is not a BCI-group for every k ≥ 3, while it is known that Zk3 is a CI-group for every 1 ≤ k ≤ 5 [32]

Some notation

Construction and basic results

Schur notation

One special case

The permutation group G is 2-closed

Schur rings