On the systems of Higman

In 1969, Higman in (Illinois J Math 13:74–80, 1969) constructed a geometry admitting HS, the non-abelian simple group discovered by Higman and Sims in (Math Z 105:110–113, 1968) [6], as its full automorphism group. One year later, Smith in (J Algebra 41:175–195, 1976) investigated the Higman systems as a natural generalization of the geometry of Higman, thus providing a classification of such systems under some additional assumption. In this paper, we complete the work started by Smith by classifying all such of Higman systems without any of the additional assumptions introduced by M. S. Smith.