A reduction theorem for primitive binary permutation groups. (Q2801723)

In this context a permutation group \((X,G)\) is called binary if the orbits of \(G\) on \(X^n\) (\(n\geq 2\)) are determined by the orbits of \(G\) on \(X^2\) in the sense that two points \(\overline x,\overline y\in X^n\) are in the same \(G\)-orbit \(\iff\) each pair of entries in \(\overline x\) is in the same \(G\)-orbit as the corresponding pair of entries in \(\overline y\).NEWLINENEWLINE \textit{G. Cherlin} [in: The Gelfand Mathematical Seminars, 1996-1999. Dedicated to the memory of Chih-Han Sah. Boston: Birkhäuser. 15-48 (2000; Zbl 0955.03040)] conjectures that if \(G\) is a finite primitive binary group, then either \(G=S_n\) acting naturally, or \(G\) is cyclic of prime order acting regularly, or \(G\) is an affine orthogonal group of the form \(V\rtimes O(V)\) where \(V\) is a vector space with an anisotropic quadratic form and \(O(V)\) is its full orthogonal group. Recent work by \textit{G. Cherlin} has established the conjecture in the case that \(G\) has an abelian socle [see J. Algebr. Comb. 43, No. 2, 339-374 (2016; Zbl 1378.20001)].NEWLINENEWLINE In the present paper the author proves the following. (Theorem A): If \(G\) is a finite primitive binary permutation group with a nonabelian socle, then either (i) \(G\) is almost simple, or (ii) \(G\) is a subgroup of a wreath product \(H~wr~S_m\) in its product action where \(H\) is a primitive binary almost simple permutation group that is not \(2\)-transitive. This implies that a proof of Cherlin's conjecture would follow from a proof of the following conjecture: the only finite primitive binary permutation groups with nonabelian socles are \(S_n\) with its natural action.