A test for commutators
Glasgow mathematical journal, Tome 17 (1976) no. 1, pp. 31-36
Voir la notice de l'article provenant de la source Cambridge University Press
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.
Liebeck, Hans. A test for commutators. Glasgow mathematical journal, Tome 17 (1976) no. 1, pp. 31-36. doi: 10.1017/S0017089500002688
@article{10_1017_S0017089500002688,
author = {Liebeck, Hans},
title = {A test for commutators},
journal = {Glasgow mathematical journal},
pages = {31--36},
year = {1976},
volume = {17},
number = {1},
doi = {10.1017/S0017089500002688},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002688/}
}
[1] 1.Macdonald, I. D., On a set of normal subgroups, Proc. Glasgow Math. Soc. 5 (1962), 137–146. Google Scholar
[2] 2.Macdonald, I. D., On cyclic commutator subgroups, J. London Math. Soc. 38 (1963), 419–422. Google Scholar | DOI
[3] 3.Rodney, D. M., Commutators and conjugacy in groups, Ph.D. thesis, University of Keele (1974). Google Scholar
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