Factorization of »-soluble and »-nilpotent Groups

then we term the elements x and y n-commutative. It is not difficult to verify that »-commutativity and (1—»)-commutativity are equivalent properties of the elements x and y, that ( — ̂ -commutativity implies ordinary commutativity, and that commuting elements are »-commutative. From any concept and property involving the fact that certain elements [or functions of elements] commute, one may derive new concepts and properties by substituting everywhere «-commutativity for the requirement of plain commutativity. This general principle may be illustrated by the following examples. n-abelian groups are groups G such that (xy)" = x'ty'' for every x and y in G. They have first been discussed by F. Levi [3 ] ; and they will play an important rôle in our discussion. Grün [2] has introduced the n-commutator subgroup. It is the smallest normal subgroup J oí G such that G/J is «-abelian ; and J may be generated by the totality of elements of the form (xy)n(xnyn)~1 with x and y in G. Dual to the «-commutator subgroup is the n-center. It is the totality of elements z in G such that (zx)n=z"xn and (xz)n = x"zn for every x in G; see Baer [l] for a discussion of this concept.