Factorization of 𝑛-soluble and 𝑛-nilpotent groups

Abstract

then we term the elements x and y n-commutative. It is not difficult to verify that n-commutativity and (1 -n)-commutativity are equivalent properties of the elements x and y, that (I)-commutativity implies ordinary commutativity, and that commuting elements are n-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 n-commutativity for the requirement of plain commutativity. This gen,eral principle may be illustrated by the following examples. n-abelian groups are groups G such that (xy)n =xnyn for every x and y in G. They have first been discussed by F. Levi [31; and they will play an important r6le in our discussion. Grun [2] has introduced the n-commutator subgroup. It is the smallest normal subgroup J of G such that G/J is n-abelian; and J may be generated by the totality of elements of the form (xy)n(xnyn)with x and y in G. Dual to the n-commutator subgroup is the n-center. It is the totality of elements z in G such that (zx) = znxn and (xz)n = xzn for every x in G; see Baer [1] for a discussion of this concept.