# 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.

• These groups are sometimes called algebra groups - this might help your search. There's a great deal of literature on their character theory. – M T Jun 18 '11 at 13:42
• thank you.do you know a paper which contains most of the known results about those groups?by the way: if someone has an interesting example of a p-group not realisable as 1+J,I would be glad if you post it. – trew Jun 18 '11 at 15:54