Gareth Tracey

Primitive amalgams and the Goldschmidt-Sims conjecture

A triple of finite groups (H, M, K), usually written H > M < K, is called a primitive amalgam if M is a subgroup of both H and K, and each of the following holds: (i) Whenever A is a normal subgroup of H contained in M, we have N_K(A)=M; and (ii) whenever B is a normal subgroup of K contained in M, we have N_H(B)=M.

Primitive amalgams arise naturally in many different contexts across pure mathematics, from Tutte's study of vertex-transitive groups of automorphisms of finite, connected, trivalent graphs; to Thompson's classification of simple N-groups; to Sims' study of point stabilizers in primitive permutation groups, and beyond. In this talk, we will discuss some recent progress on the central conjecture from the theory of primitive amalgams: the Goldschmidt--Sims conjecture. Joint work with L. Pyber.