p-groups realisable as 1+J,where J is a nilpotent finite F-Algebra There is a functor F:   {finite nilpotent Algebra over a finite field F}  -> {finite p-groups}
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 p-groups 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.
 A: For algebras over non-prime fields, a good reference is Isaacs (1995).  It proves a fundamental theorem establishing that these F-algebra groups behave like |F|-groups, not just like p-groups, for char(F) = p.  I think these groups have been studied over Z/pZ or even Z/pnZ for a very long time.
As far as a p-group not of the form 1 + J: any cyclic p-group 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 2-subgroup of GL(4,2), and hence has exponent dividing 4.  Similar bounds hold for odd p: the cyclic group of order p2 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 p-restricted Lie algebras in finite p-groups.  This takes a p-group 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 p-groups" should work).


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*Isaacs, I. M.
Characters of groups associated with finite algebras. 
J. Algebra 177 (1995), no. 3, 708–730.
MR1358482
DOI:10.1006/jabr.1995.1325
