Cyclic Subgroup Lattices as Universal Sources of Power-Type Graphs
Mahsa Mirzargar, Sezer Sorgun, Mohammad Javad Nadjafi Arani
TL;DR
The paper tackles how power-type graphs encode finite-group structure by establishing a bidirectional, purely combinatorial correspondence between the enhanced power graph $\mathrm{EPow}(G)$ and the cyclic subgroup lattice $\mathcal{L}_c(G)$. It develops a bottom-up reconstruction framework that recovers $\mathrm{EPow}(G)$ from $\mathcal{L}_c(G)$ and, in turn, derives $\mathrm{Pow}(G)$, $\overrightarrow{\mathrm{Pow}}(G)$, and $\mathrm{D}(G)$ from the lattice, thereby presenting a unified lattice-centric view of these graphs. A key contribution is a canonical labeling of elements via generators of cyclic subgroups, enabling an operation-free, graph-to-group transfer that also reduces the isomorphism problem to isomorphism of $\mathcal{L}_c(G)$. The results pave the way for graph-theoretic and lattice-theoretic analyses of finite groups and suggest computational applications for group-invariant reconstruction and classification based on purely combinatorial data.
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.