On the directed normalizing graph associated with a group
Costantino Delizia, Michele Gaeta, Carmine Monetta
TL;DR
The paper introduces the directed normalizing graph $\vec{\Gamma}_{\rm norm}(G)$, where an edge $x\to y$ encodes that $\langle x\rangle$ is normal in $\langle x,y\rangle$, and studies the induced graph after removing bidirectional universal vertices. It develops a detailed structural analysis of universal vertices, including the Baer norm characterization, and then delivers complete and partial classifications for when the associated undirected graph is complete, deriving implications for nilpotency and Dedekind groups. The core results center on the connectivity and diameter of the residual graph $\vec{\Delta}_{\rm norm}(G)$, with sharp criteria for soluble groups with trivial center: strongly disconnected exactly for Frobenius and $2$-Frobenius groups, and precise diameter bounds (up to $8$ in general, with tighter bounds under additional hypotheses). The work further connects these graph-theoretic properties to classical group invariants such as the Fitting subgroup and the Baer norm, enriching the interplay between group structure and associated normalizing graphs.
Abstract
In this paper we investigate the $directed$ $normalizing$ $graph$ associated with a group $G$, defined as the simple directed graph whose vertices are the elements of $G$, with an arrow from $x$ to $y$ whenever the subgroup $\langle x \rangle$ is normal in $\langle x, y \rangle$. Our analysis focuses on the set of bidirectional universal vertices and, in particular, on the induced subgraph obtained by removing them, where the most interesting connectivity phenomena occur. We characterize the groups for which this induced subgraph is strongly connected and determine bounds for its diameter. Finally, we show how properties of this graph reflect algebraic features of the underlying group.