The Guralnick-Thompson Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $\sigma\in S_n$ let $\operatorname{ind}(\sigma)=n-c(\sigma)$, where $c(\sigma)$ counts all permutation cycles, including fixed points. A genus-zero system is a tuple $(\sigma_1,\ldots,\sigma_r)$ generating a transitive subgroup $G\leq S_n$ and satisfying \[\sigma_1\cdots\sigma_r=1,\qquad\sum_{i=1}^r\operatorname{ind}(\sigma_i)=2(n-1).\] Composition factors are the simple quotients in a composition series of $G$, considered up to group isomorphism. Let $E^*(0)$ be the union of these isomorphism types over all genus-zero systems, after excluding the alternating groups $A_m$ for $m\geq5$ and the cyclic groups $C_p$ of prime order.

Statement recorded in the supplied source.

Classify the simple composition factors of all finite groups generated by genus-zero systems; equivalently, determine the exceptional set $E^*(0)$ in addition to the alternating and prime-order cyclic families. The named Guralnick--Thompson conjecture asserts finiteness of the analogous exceptional set at each fixed genus, defined by replacing $2(n-1)$ by $2(n+g-1)$; the cited source records that finiteness theorem as established. [S2]

Short English statement

Determine which finite simple groups occur as composition factors of genus-zero monodromy groups. These are groups generated by transitive permutation systems satisfying the genus-zero relation, with a historically proved finite-exception principle.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1189 (ALG-025). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Daniel Frohardt, Robert Guralnick and Kay Magaard, Primitive Monodromy Groups of Genus at most Two, arXiv:1401.1303 (2014). Introduction and Definition1, pages1--2. https://arxiv.org/pdf/1401.1303.