On finite nilpotent groups with the same enhanced power graph

TL;DR

The paper investigates how the enhanced power graph $P_e(G)$ constrains the structure of a finite group, focusing on nilpotent groups. It proves a structural reduction: if $P_e(G)\cong P_e(H)$ for nilpotent $G$, then $H$ is nilpotent and, for each prime $p$, the Sylow $p$-subgroups satisfy $P_e(G_{P})\cong P_e(H_{Q})$, linking global isomorphisms to prime-power components. A counterexample shows $P_e$-isomorphism need not determine the group in general, but the main theorem yields a decomposition principle that underpins classifications of nilpotent groups with uniquely determined $P_e$ (and, symmetrically, $Pow$ and $C$). The paper further identifies explicit families of nilpotent groups (e.g., $Q_8\times \mathbb{Z}_n$ with odd $n$, $(\mathbb{Z}_2)^m\times \mathbb{Z}_n$, and $\mathbb{Z}_p\times \mathbb{Z}_p\times \mathbb{Z}_n$ with $(n,p)=1$) for which the enhanced power graph uniquely determines the group, and extends these results to the related power and cyclic graphs, offering a unified perspective on graph-based group identification.

Abstract

The enhanced power graph of a group $G$ is the graph $P_e(G)$ whose vertex set is $G$, such that two distinct vertices $x$ and $y$, are adjacent if $\langle x, y\rangle$ is cyclic. In this paper, we analyze the structure of the enhanced power graph of a finite nilpotent group in terms of the enhanced power graphs of its Sylow subgroups. We establish that for two nilpotent groups, their enhanced power graphs are isomorphic if and only if the enhanced power graphs of their Sylow subgroups are isomorphic. Additionally, we identify specific nilpotent groups for which the enhanced power graphs uniquely characterize the group structure, meaning that if $P_e(G)\cong P_e(H)$ then $G \cong H$. Finally, we extend these results to power graphs and cyclic graphs.

Figures (4)

• Figure 1: $P_{e}(\mathbb{Z}_3\times \mathbb{Z}_3\times \mathbb{Z}_3)\cong P_{e}(S)$
• Figure 2: The enhanced graph of $Q_8\times \mathbb{Z}_{n}$ where $n$ is an odd integer, consist of three complete graphs $K_{2n}$ with $2n$ dominating vertices.
• Figure 3: The enchanced graph of $(\prod_{i=1}^m \mathbb{Z}_2)\times \mathbb{Z}_n$, where $n$ is an odd integer, consist of $2^m-1$ complete graphs $K_{n}$ with $n$ dominating vertices.
• Figure 4: The enchanced graph of $\mathbb{Z}_p\times \mathbb{Z}_p\times \mathbb{Z}_{n}$, where $gcd(n,p)=1$, consist of $p+1$ complete graph $K_{2n}$ with $n$ dominating vertices.

Theorems & Definitions (8)

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