Dimension subgroups andp-th powers inp-groups
- 54 Downloads
We prove that if the nilpotence class of ap-group is strictly less thanp kthen every product ofp k-thpowers can be written as thep-th power of an element. Scoppola and Shalev have proven the same thing for groups of class strictly less thanp k−p k−1. They also provide an example which proves that ours is the best possible result. This is a generalization of the well known fact that in groups of class strictly less thanp every product ofp-powers is again ap-th power. Along the way we prove results of independent interest on dimension subgroups ofp-groups.
KeywordsPositive Integer Normal Subgroup Repeated Application Nilpotence Class Lower Central Series
Unable to display preview. Download preview PDF.
- [DdMS]J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal,Analytic Pro-p Groups, 2nd edition, Cambridge Studies in Advanced Mathematics, 61, Cambridge University Press, 1999.Google Scholar
- [GJ]J. González and A. Jaikin,On the structure of normal subgroups of potent p-groups, submitted.Google Scholar
- [W2]L. Wilson,The power-commutator structure of certain finite p-groups, Journal of Group Theory, to appear.Google Scholar
- [W3]L. Wilson,Powerful subgroups of 2-groups, submitted.Google Scholar