Powerfully Nilpotent Groups of Class 2

Abstract

In this article, we investigate the powerful nilpotency class of powerfully nilpotent groups of standard nilpotency class 2. We outline the process of collecting data using the computer algebra system GAP, formulating a conjecture based on the data, and finally we prove the conjecture. In particular, we prove that for a powerfully nilpotent group of nilpotency class 2 and order pⁿ, where p is an odd prime, the powerful nilpotency class of G is at most the integer part of $n/2$. We also identify and explain what this means in terms of the powerful coclass of the group.

Keywords:

• powerfully nilpotent
• coclass
• powerful p-group

2010 Mathematics Subject Classification:

• 20D15

Declaration of interest

No potential conflict of interest was reported by the author(s).

Acknowledgments

The author is grateful to Gunnar Traustason and Gareth Tracey for their suggestions and advice with this article. The author is especially grateful for the suggestions of Gunnar Traustason with regard to the proof of Theorem 21.

Additional information

Funding

This work was completed during the author’s PhD at the University of Bath, under the supervision of Professor Gunnar Traustason, funded by the EPSRC (grant number 1652316).