In this paper we continue developing the theory of symplectic\ud alternating algebras that was started in Traustason (2008). We focus on nilpotency, solubility and nil-algebras. We show in particular that symplectic alternating nil-2 algebras are always nilpotent and classify all nil-algebras of dimension up to 8

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We develop a structure theory for nilpotent symplectic alternating algebras.

This paper begins the development of a theory of what we will call symplectic alternating algebras. They have arisen in the study of 2-Engel groups but seem also to be of interest in their own right. The main part of the paper deals with the challenging classification of some algebras of this kind which arise in the context of 2-Engel groups and give some new information about these groups. The main result is that there are 31 such algebras with dimension 6 over the field of three elements.

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.