On the Symmetric Normaliser Graph of a Group

TL;DR

This paper introduces the symmetric normaliser graph of a group, exploring its properties, hierarchical position among related graphs, and conditions for equality with other well-known group graphs.

Contribution

It defines the symmetric normaliser graph, analyzes its placement in the graph hierarchy of groups, and establishes conditions for its equivalence with other key group graphs.

Findings

The symmetric normaliser graph refines the existing hierarchy of group graphs.

Conditions are provided for when it equals the commuting or nilpotent graph.

Its edges are related to the Engel and non-generating graphs.

Abstract

In this paper we introduce the symmetric normaliser graph of a group $G$. The vertex set of this graph consists of elements of the group. Vertices $x$ and $y$ are adjacent if $x$ lies in the normaliser of $\langle y \rangle$ and $y$ lies in the normaliser of $\langle x \rangle$. We investigate the hierarchical position this graph occupies in the hierarchy of graphs defined on groups. We show that the existing hierarchy is further refined by this graph and that the edges of this graph lie between the edges of the commuting graph and the nilpotent graph. For finite groups, we prove a necessary and sufficient condition for the symmetric normaliser graph to be equal to the commuting graph and similarly, for equality with the nilpotent graph. The edge set of the symmetric normaliser graph is also a subset of the edge set of the Engel graph of a group and has connections to the non-generating graph of a group.