A classification of nilpotent 3-BCI groups

TL;DR

This paper classifies finite nilpotent groups that ensure all bi-Cayley graphs of valency up to 3 are BCI-graphs, revealing a specific structural characterization involving homocyclic groups and certain 2-groups.

Contribution

It provides a complete characterization of nilpotent 3-BCI-groups, identifying their precise algebraic structure.

Findings

Finite nilpotent groups are 3-BCI if and only if they are of the form U × V with specified properties.

The groups U are homocyclic of odd order, V is trivial or one of the specified 2-groups.

This classification extends understanding of symmetry properties of bi-Cayley graphs.

Abstract

Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph $\bcay(G,S)$ is the graph whose vertex set is $G \times \{0,1\}$ and edge set is $\{\{(x,0),(s x,1)\} : x \in G, s\in S \}$. A bi-Cayley graph $\bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph $\bcay(G,T),$ $\bcay(G,S) \cong \bcay(G,T)$ implies that $T = g S^\alpha$ for some $g \in G$ and $\alpha \in \aut(G)$. A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form $U \times V,$ where $U$ is a homocyclic group of odd order, and $V$ is trivial or one of the groups $\Z_{2^r},$ $\Z_2^r$ and $\Q_8$.