Conditions of finiteness on sharply 2-transitive groups

Abstract

A well known theorem of H. Zassenhaus [6], which also appears in M. Hall [1], p. 382, states that a finite sharply 2-transitive permutation group 1) is isomorphic to the group of linear transformations x--', a + m . x on a finite near-field. M ore generally, one can show that the group of linear transformations on an algebraic structure called a near-domain (see Definition A) is sharply 2-transitive and that, up to isomorphism as permutation groups, each sharply 2-transitive group is isomorphic to the group of linear transformations on a uniquely determined near-domain, [2], [3], [4]. Hence the class of sharply 2-transitive groups is completely characterized by the class of neardomains. To the authors' knowledge, the question as to the existence of near-domains which are not near-fields is open. Some results on this question are given in [4]. In this paper, a theorem is proved which states that a near-domain is a near-field if and only if a certain subset is finite. Thus the theorem of Zassenhaus which states, in the terminology used here, that every finite near-domain is a near-field, can be generalized to more relaxed conditions of finiteness (Theorem A, Coroliaries 1 and 2). The latter two results are then interpreted in terms of sharply 2-transitive groups (Corollary 3).