The Solution of Boen's Problem

A number of authors have proved special cases of Ci as well as special cases of more general propositions [l, 2, 3, 5, 6, 7, 8] . Both Ci and all of the generalizations of it which have been considered in the literature follow from Theorem 1 which appears below. In [2] it is observed that if P is a smallest counterexample to Ci, then there is associated with P , an anticommutative (not necessarily associative) algebra A over GF(p), whose dimension coincides with the number of elements in a minimal generating set of the £-automorphic group P. Further, if G is the hypothesized group of automorphisms of P , then G also acts as a group of automorphisms of A in such manner that both A and the Frattini-factor group of P are isomorphic as GF(p)G-modules. Accordingly, Kostrikin [6] has introduced the notion of homogeneous algebra, i.e. a finite dimensional algebra A over a finite field GF(g), which admits a group of automorphisms G, transitively permuting its nonzero elements. Such algebras enjoy two basic properties: (Pi) if q is odd, they are anticommutative [6], and (P2) left multiplication by an element induces a nilpotent transformation of A [2]. Then Ci is a consequence of the proposition :