A test for commutators

Abstract

In the course of a study of commutator subgroups I. D. Macdonald [1] presented the free nilpotent group G4 of class 2 on 4 generators as an example of a nilpotent group whose commutator subgroup has elements that are not commutators. To demonstrate this he proceeded as follows: let G4 = 〈a1, a2, a3, a4〉 and put cij = [ai, aj] for 1 ≦ i < j ≦ 4. Then the relations in G4 are [cij, ak] = 1 for 1 ≦ i < j ≦ 4 and 1 ≦ k ≦ 4, and their consequences. Macdonald observed that an arbitrary commutator may be written aswhich simplifies towhere δij = αiβj - αjβi The indices δij satisfy the relationIt follows that the element c13c24 in G′4 (for which δ12 = δ14 = δ23 = δ34 = 0 and δ13 = δ24 = 1) is not a commutator.