There is a functor F: {finite nilpotent Algebra over a finite field F} > {finite pgroups} sending J to 1+J.(You can think of J as the Jacobsonradical of 1F+J) and sending f:A>B to F(f):1+A > 1+B with F(f)(1+a)=1+f(b). I want to know which classes of pgroups are realisable as 1+J and maybe if you see some interesting properties of the functor F. For example if J^2 = 0,then G=1+J is elementary abelian and if J^3=0 then G=1+J is special. Such realisations could be interesting since,we have for example in general G' $\leq $ 1+J^2 and a series 1+J $\leq$ 1+J^2 ... $\leq$ 1+J^k =0 and there it is known that the characters of 1+J are induced from linear characters of a subgroup 1+A where A is a subalgebra of J.

For algebras over nonprime fields, a good reference is Isaacs (1995). It proves a fundamental theorem establishing that these Falgebra groups behave like Fgroups, not just like pgroups, for char(F) = p. I think these groups have been studied over Z/pZ or even Z/p^{n}Z for a very long time. As far as a pgroup not of the form 1 + J: any cyclic pgroup of large enough exponent will do. For instance, the cyclic group of order 8 is not of the form 1 + J: P = J, and so J has dimension 3 over F = Z/2Z. Hence 1F+1J has dimension 4, and P is contained in a Sylow 2subgroup of GL(4,2), and hence has exponent dividing 4. Similar bounds hold for odd p: the cyclic group of order p^{2} is not an algebra group, for instance. Every group of order dividing 8 (other than the cyclic group of order 8) is an algebra group. I assume you have already seen the use of prestricted Lie algebras in finite pgroups. This takes a pgroup and associates a Lie algebra with pretty similar properties as J has in 1+J. If not, then this may be exactly what you are looking for and is very well studied (refs on requests, but any "good book on pgroups" should work).


