On Shumyatsky’s Question on Coprime Commutators: Nineteen Sporadic Groups, the Tits Group and All Simple Groups of Order Less Than 10⁹

Shumyatsky asked whether each element of a nonabelian finite simple group can be written as \([x,y]\) with \(x\) and \(y\) of coprime orders (Kourovka Notebook, Problem 18.117). The answer is known to be yes for the alternating groups (Shumyatsky), for the groups \(\mathrm{PSL}(2,q)\) (Pellegrini and Shumyatsky) and for the Suzuki groups (Zini); without the condition on the orders the statement is Ore's conjecture, proved by Liebeck, O'Brien, Shalev and Tiep. We give a partial answer; the proofs are computer-assisted. In the sporadic groups \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\), \(M_{24}\), \(J_1\), \(J_2\), \(J_3\), \(\mathrm{HS}\), \(\mathrm{McL}\), \(\mathrm{He}\), \(\mathrm{Ru}\), \(\mathrm{Suz}\), \(\mathrm{Co}_3\), \(\mathrm{Co}_2\) and \(\mathrm{Fi}_{22}\), in the Tits group and in 36 simple groups of Lie type, every element is a commutator \([x,y]\) with \(x\) a \(2\)-element and \(y\) of odd order; the certificate is one stored pair per conjugacy class, re-checked by exact conjugacy tests in GAP. In \(\mathrm{O'N}\), \(\mathrm{Ly}\) and \(J_4\) every element is a commutator of two elements of coprime orders, by short lemmas applied to the character tables of the GAP Character Table Library and explicit witnesses in five constructed subgroups, every step through a subgroup being checked against all possible class fusions. Consequently every nonabelian finite simple group of order less than \(10^9\) has the property, and so do 19 of the 26 sporadic groups. A withdrawn preprint of Lucchini and Spiga had reported a computer search, without published witnesses, for 19 groups, among them \(\mathrm{O'N}\) and \(\mathrm{Fi}_{23}\). The question remains open: for \(\mathrm{HN}\), \(\mathrm{Th}\), \(\mathrm{Co}_1\), \(\mathrm{Fi}_{23}\), \(\mathrm{Fi}_{24}'\), \(B\) and \(M\) only some of the classes are certified here, and the groups of Lie type of order at least \(10^9\), apart from \(\mathrm{PSL}(2,q)\), the Suzuki groups and five groups of the list above, are not treated. The results rest on GAP's exact algorithms, on the library character tables and on the ATLAS lists of maximal subgroups. This is an unrefereed note.