On the Symmetric Normaliser Graph of a Group

TL;DR

The paper defines the symmetric normaliser graph $SNorm(G)$ on a group $G$ and places it within the established graph hierarchy on groups. It proves edge-set inclusions $E(Com(G))\subseteq E(SNorm(G))\subseteq E(Nilp(G))$, with further relations to $Engel(G)$ and $NGen(G)$, and provides finite-group criteria for $SNorm(G)=Com(G)$ and $SNorm(G)=Nilp(G)$. It then uses SNNC and exponent-critical $p$-groups to characterize when $Com(G)$ and $SNorm(G)$ differ, showing that $Com(G)=SNorm(G)$ iff $G$ has no SNNC-subgroup. Finally, it analyzes completeness and equality with $Nilp$, $EPow$, and $Pow$ via Sylow-subgroup structure and highlights open questions.

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.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof