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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.

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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
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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).

  • 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
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The groups of order 16 that are not algebra groups are exactly the cyclic, the maximal class (D16, QD16, Q16), and the modular group. The groups of order p^3 (p odd) that are not algebra groups are the cyclic and the extra-special group of exponent p^2. I am nervous about p^4: it seems (too) rare to be an algebra group. –  Jack Schmidt Jun 19 '11 at 14:22
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