A CLASSIFICATION OF SOME INSOLUBLE VARIETIES OF GROUPS OF EXPONENT-4

We classify the subvarieties of the variety B-4 boolean AND C-4 of all 4-Engel groups of exponent 4 which contain the proper subvariety defi ned by M. Quick (J. London Math. Soc, to appear), and from this we ded uce a classification of all varieties of groups of exponent 4 which co ntain this subvariety. We also use the first classification to obtain a reduction for the finite basis problem for varieties of groups of ex ponent 4. (C) 1997 Academic Press.