Kourovka Notebook Problem 21.94

Definitions and mathematical statement

For a finite group $G$, let $\pi(G)$ be the set of primes dividing $|G|$. Its Gruenberg--Kegel graph, or prime graph, has vertex set $\pi(G)$ and an edge between distinct primes $p,q$ exactly when $G$ contains an element of order $pq$. There are no loops or multiple edges.

Write $\overline{\operatorname{GK}}(G)$ for the abstract graph obtained by forgetting which prime labels each vertex. An isomorphism of such graphs is any adjacency-preserving bijection between their vertex sets; it need not preserve prime labels. Call $G$ recognizable by its unlabelled prime graph if \[ \forall\text{ finite groups }H,\qquad \overline{\operatorname{GK}}(H)\cong \overline{\operatorname{GK}}(G)\ \Longrightarrow\ H\cong G. \] The question asks whether infinitely many pairwise nonisomorphic finite groups have this property. Equivalently, are there groups $G_1,G_2,\ldots$ with $G_i\not\cong G_j$ for $i\ne j$, each uniquely determined among all finite groups by the abstract isomorphism type of its prime graph?

The possible competitors $H$ are unrestricted finite groups: they are not assumed to have the same order, the same prime divisors, or the same composition factors as $G$. In particular the recognition condition must exclude a competitor whose prime graph has the same shape but different vertex labels. Group isomorphism, rather than equality of presentations, is the required conclusion.

Short English statement

Are infinitely many finite groups uniquely determined, among all finite groups, by the shape of their unlabelled prime graphs?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2603 (KOU-21.94), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.94. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.