A FREEDOM THEOREM FOR GROUPS WITH ONE DEFINING RELATION IN THE VARIETIES OF SOLVABLE AND NILPOTENT GROUPS OF GIVEN LENGTHS

Abstract

The freedom theorem of Magnus [1] is well known: if a group G is given by generators x1, x2,?... and a single defining relation r?=?1, and if r is not conjugate to any word in x2,?..., then the elements x2,?... freely generate in G a free subgroup. In this note analogous theorems of Magnus are established for groups given by one defining relation in the varieties of soluble and nilpotent groups of given lengths. Bibliography: 7 titles.