Kourovka Notebook Problem 19.52

Definitions and mathematical statement

For a finite group $G$, let $\pi(G)$ be the set of prime divisors of $|G|$. Its Gruenberg--Kegel graph (prime graph) $GK(G)$ is the simple undirected graph with vertex set $\pi(G)$, in which distinct primes $p,q$ are adjacent if and only if $G$ contains an element of order $pq$. A $3$-coclique is a set of three distinct vertices, no two of which are adjacent. A group $H$ is solvable if its derived series, defined by $H^{(0)}=H$ and $H^{(i+1)}=[H^{(i)},H^{(i)}]$, reaches the trivial group after finitely many steps.

Does there exist a finite nonsolvable group $G$ such that \[ \begin{gathered} \text{for every three distinct }p,q,r\in\pi(G),\text{ at least one of }pq,pr,qr\\ \text{is the order of an element of }G, \end{gathered} \] and \[ \text{for every finite solvable group }H,\qquad GK(G)\not\cong GK(H)? \] Here graph isomorphism means an isomorphism of abstract graphs; it need not preserve the prime labels of vertices.

The supplied source further states that no such example exists among almost simple groups. A finite group $A$ is almost simple if $S\leq A\leq\operatorname{Aut}(S)$ for some finite nonabelian simple group $S$, identified with its inner automorphism group. This exclusion is retained as supplied background; the record gives no separate theorem reference or proof for it, and it has not been independently verified here.

Short English statement

Does there exist a finite nonsolvable group whose prime graph has no three pairwise nonadjacent vertices and is not isomorphic to the prime graph of any finite solvable group? The supplied record labels the problem open and states that almost simple groups provide no examples; these are supplied source assessments, not independent verifications.

Sources

• Local UnsolvedMath registry, permanent ID $701600$, exact source problem code catalogue entry, title Kourovka Notebook Problem 19.52; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry record was created and updated on 2026-10-16T14:00:00Z and labels the problem open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 19.52, proposed by N. V. Maslova in 2018. The supplied background identifies the 19th issue (2018) and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The assertion excluding almost simple groups is part of the supplied statement. No separate bibliographic reference for that assertion is supplied. The notebook citation and registry status have not been independently checked.