Cyclic Subgroup Lattices as Universal Sources of Power-Type Graphs

TL;DR

This paper establishes a complete combinatorial correspondence between power-type graphs of finite groups and their cyclic subgroup lattices, enabling mutual reconstruction and offering a new framework for group analysis.

Contribution

It introduces a duality between power-type graphs and cyclic subgroup lattices, allowing for their mutual reconstruction and a purely combinatorial approach to studying finite groups.

Findings

Enhanced power graph determines the cyclic subgroup lattice.

Power graph, directed power graph, and difference graph can be reconstructed from the lattice.

Provides a new combinatorial framework for analyzing finite groups.

Abstract

Power-type graphs, such as the power graph, the directed power graph, the enhanced power graph and the difference graph, encode significant information about the internal structure of a finite group. Despite substantial investigation in recent years, the precise relationship between these graphs and the subgroup lattice of the underlying group has remained only partially understood. In this paper we establish a complete, explicit, and purely combinatorial correspondence between the enhanced power graph and the lattice of cyclic subgroups $\mathcal{L}_c(G)$ of a finite group $G$. We prove that these two objects determine each other uniquely: an unlabeled enhanced power graph suffices to reconstruct $\mathcal{L}_c(G)$, and conversely, the labeled enhanced power graph can be reconstructed directly from $\mathcal{L}_c(G)$. Exploiting this duality, we demonstrate that the reconstruction principle applies equally to the power graph, the directed power graph, and the difference graph, which may all be derived solely from the cyclic subgroup lattice, independent of the group operation. This bidirectional correspondence yields a combinatorial equivalence between power-type graphs and the cyclic subgroup lattice, providing a new framework for analyzing finite groups through graph-theoretic and lattice-theoretic data, free from algebraic complexity.