MIXED IDENTITIES AND MIXED VARIETIES OF GROUPS

A mixed identity in variables over a group is a word (where the coefficients lie in , , and ) taking the value 1 for any values of the variables in . The concept of a mixed variety of groups is introduced as an object corresponding to a certain set of mixed identities and generalizing the concept of a variety of groups; an analogue of Birkhoff's theorem is proved; minimal mixed varieties generated by a finite group are described; the question of whether the mixed identities of a group can be derived from its identities is studied; and for nilpotent and metabelian groups it is established that all their mixed identities with coefficients in a finitely generated subgroup are finitely based, from which the same property is deduced for the identities of such groups with finitely many distinguished points.Bibliography: 16 titles.