Association schemes and permutation groups (Q1810629)

A coherent algebra is an algebra of \(n\times n\) complex matrices which has a basis \(\{B_0,B_1,\dots,B_t\}\) consisting of matrices with entries 0 and 1 satisfying the following conditions: (a) \(B_0+B_1+\cdots+B_t=J_n\); (b) there is a subset of \(\{B_0,B_1,\dots,B_t\}\) with sum \(I_n\); (c) the set \(\{B_0,B_1,\dots,B_t\}\) is closed under transposition. The algebra is called homogeneous if the subset referred to in (b) contains just one element, which we take to be \(B_0=I_n\). We call a transitive permutation group AS-free if it preserves no non-trivial association scheme. A transitive permutation group \(G\) is called AS-friendly if there is a unique finest \(G\)-invariant association scheme. Primitive groups which are not AS-friendly do exist; the smallest example is \(\text{PSL}(2,13)\), acting on 78 points (Cameron). There are AS-free groups which are not stratifiable: \(\text{PSL}(3,3)\), degree 234; \(M_{12}\), degree 1320; \(J_1\), degree 1463, 1540 or 1596; and \(J_2\), degree 1800 (Soicher). Corollary 6. If \(G\) has a regular Abelian subgroup \(A\), then \(G\) is stratifiable; and if \(G\) contains the holomorph of \(A\), then \(G\) is geneously transitive. For a finite group \(G\), the following conditions are equivalent: (1) the regulear action of \(G\) is AS-friendly; (2) the regular action of \(G\) is stratifiable; (3) either \(G\) is Abelian or \(G\simeq Q_8\times A\), where \(Q_8\) is the quaternion group of order 8 and \(A\) is an elementary Abelian 2-group (Theorem 7). Theorem 9. Let \(G\) be an AS-free permutation group. Then \(G\) is primitive and basic. Moreover, \(G\) is 2-homogeneous, of diagonal type, or almost simple. Among other things it is shown that a diagonal group with non-Abelian base group cannot be stratifiable if it has ten or more factors, or generously transitive if it has nine or more.