ON A CONNECTION BETWEEN HUGHES' CONJECTURE AND RELATIONS IN FINITE GROUPS OF PRIME EXPONENT

Abstract

Under the assumption that the ideal of relations of a?free 3-generator group of period? does not coincide modulo? with the -Engel ideal it is proved that there exist -groups? of nilpotence degree in which the index of the Hughes subgroup is (Theorem?1). The author also finds that Macdonald's result on -groups of class is best possible (at least for ). The proof is based on direct computations almost the same as in work of A.I.?Kostrikin dating from 1957; it uses properties of the coefficients in the Baker-Hausdorff formula. An automorphism? of order? of the group? is called splitting if for all? in?. It is easy to see that if and only if , where? is a?splitting automorphism of order? of . It is proved that if a?finite -group? admits a?splitting automorphism? of order? and the nilpotency degree of does not exceed , then? is regular (Theorem?2). From Theorem?2 it is possible to deduce an independent proof of Hughes' conjecture for -groups of class . On the basis of Theorem?1 the author constructs examples of -groups admitting a?splitting automorphism of order? for which the associated Lie ring is not a?-Engel ring.Bibliography: 12 titles.