On the automorphism groups of rank-4 primitive coherent configurations

Abstract

The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree ≥ cn for some constant c > 0. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3.In this paper, we extend Babai's result to primitive coherent configurations of rank 4, confirming the conjecture in this special case. The proofs combine structural and spectral methods.

Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed

“…(8) An important direction of study is the extension of known results about the minimal degree of primitive groups (often obtained via CFSG) to the motion of strongly regular graphs, distance-regular graphs, and primitive coherent configurations (PCCs). Some work in this direction has already been done, see, e. g., [Ba81,Ba15], and the profound results in [SW15, Ki21a,Ki21b,Ki21c]. A conjecture of this author that motivates Kivva's work [Ki21a,Ki21b,Ki21c] is the following.…”

“…Some work in this direction has already been done, see, e. g., [Ba81,Ba15], and the profound results in [SW15, Ki21a,Ki21b,Ki21c]. A conjecture of this author that motivates Kivva's work [Ki21a,Ki21b,Ki21c] is the following.…”