The groups with order [formula omitted] for odd prime p

Abstract

We determine product presentations for the nilpotent Lie rings with order p 7 where p ⩾ 7 is prime, and then use the Baker–Campbell–Hausdorff formula to construct power-commutator presentations for the corresponding groups. The number of such groups is a polynomial depending on p whose leading term is 3 p 5 . We complete the determination of groups with order p 7 for p = 3 , 5 using the p-group generation algorithm. We provide access to the resulting presentations for the groups via a database distributed with computer algebra systems.