Kourovka Notebook Problem 20.87

Definitions and mathematical statement

For a finite group $G$ and $g\in G$, let $o(g)$ denote the order of $g$, and define \[ \pi(o(g))=\{p\in{\mathbb{N}} : p\text{ is prime and }p\mid o(g)\}. \] The condition in the question is \[ \forall g\in G,\qquad |\pi(o(g))|\leq 2. \] Thus the bound concerns distinct prime divisors; their exponents are unrestricted, and the identity has no prime divisors in its order.

A composition series of $G$ is a chain \[ 1=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_n=G \] with $n\geq 0$, such that each quotient $G_i/G_{i-1}$, for $1\leq i\leq n$, is a nontrivial simple group. These quotients are the composition factors.

Determine, up to isomorphism, all nonabelian finite simple groups $S$ for which there exist a finite group $G$ satisfying the displayed element-order condition, a composition series of $G$, and an index $i\in\{1,\ldots,n\}$ such that \[ G_i/G_{i-1}\cong S. \] The supplied source points separately to a reference for the case in which every element order has at most one prime divisor; it supplies no theorem from that reference. The local record labels this classification problem open; that label is supplied source metadata, not an independently verified literature assessment.

Short English statement

Classify the nonabelian simple groups that occur as composition factors of finite groups in which every element order has at most two distinct prime divisors. The question is attributed to W. J. Shi (2022). The supplied record labels it open and gives a separate reference for the one-prime case.

Sources

• Local UnsolvedMath source registry, permanent ID $701745$, exact source problem code catalogue entry, title Kourovka Notebook Problem 20.87. The supplied record labels the problem open and was updated on 2026-10-16.
• W. J. Shi, proposed in 2022, Problem 20.87 in The Kourovka Notebook (Unsolved Problems in Group Theory). The supplied background cites the 20th issue (2022) and the 21st issue edition (2026), with identifier arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Reference supplied specifically for the one-prime case: https://arxiv.org/pdf/2003.09445.pdf. No authors, title, or precise result from this reference are included in the supplied record.