Powerfully nilpotent groups

Abstract

We introduce a special class of powerful p-groups that we call powerfully nilpotent groups that are finite p-groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that leads naturally to a classification in terms of an 'ancestry tree' and powerful coclass. We show that there are finitely many powerfully nilpotent p-groups of each given powerful coclass and develop some general theory for this class of groups. We also determine the growth of powerfully nilpotent groups of exponent p 2 and order p n where p is odd. The number of these is f (n) = p αn 3 +o(n 3 ) where α = 9+4 √ 2 394 . For the larger class of all powerful groups of exponent p 2 and order p n , where p is odd, the number is p 2 27 n 3 +o(n 3 ) . Thus here the class of powerfully nilpotent p-groups is large while sparse within the larger class of powerful p-groups.

Powerfully nilpotent groups of rank 2 or small order

“…Now the possible types for non-abelian groups of rank greater than 2 are (1, 1, 1, 1, 2), (1, 1, 1, 3), (1, 1, 2, 2), (1,1,4), (1,2,3) and (2,2,2). Those that are not covered by the general examples above are (2, 2, 2), (1,2,3) and (1,1,2,2). Let G be a group of type (2,2,2).…”

“…, a r such that G = a 1 · · · a r and such that |G| = o(a 1 ) · · · o(a r ). In [3,Theorem 2.7] we furthermore showed that the generators can be chosen such that the following chain is powerfully central G ≥ a p 1 , a 2 , . .…”

“…Definition. A powerful p-group G is powerfully nilpotent if it has an ascending chain of subgroups of the form {1} = H 0 A natural approach is to develop something that corresponds to a coclass theory for finite p-groups where coclass is replaced by powerful coclass and in [3] we proved that there are indeed, for any fixed prime p, finitely many powerfully nilpotent p-groups of any given powerful coclass. More precisely, if G is a powerfully nilpotent p-group of rank r and exponent p e then we showed that r ≤ n − c + 1 and e ≤ n − c + 1.…”

Powerfully nilpotent groups of maximal powerful class

“…We conjectured in [6] that the bound given in (b) is attained and we will show in this paper that this is the case when p > r . A powerfully nilpotent group, for which this bound is attained, will be called a group of maximal powerful class.…”

“…Theorem [6] Let G be a powerfully nilpotent group of rank r ≥ 2 that has a maximal tail. Suppose G has order p n , powerful class c and exponent p e .…”