I’m trying to copy the following construction of P. Forster in group theory for Lie algebras. He takes a non-abelian simple group E which has an FpE-module V such that R = Rad(V ) is faithful and irreducible as an FpE-module and V/R is an irreducible trivial FpE-module. Let H be the semidirect product V ⋋ E. Then H′ = RE (here the dash denotes the derived subgroup) is a primitive group and |H : H′ | = p. Next he takes an FqH-module W with centralizer equal to H′ where q \neq p. He then forms the semidirect product W ⋋H. He doesn’t appear to say explicitly which group E is. Does anyone know of a simple Lie algebra over a field of characteristic p for which a similar construction may be possible?
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$\begingroup$ I don't know what it means in terms of Lie algebras, but that construction works for any choice of finite non-Abelian simple group $E$ whose order is divisible by $p$. The key point is that the projective cover of the trivial module has a non-trivial (hence faithful) simple module as a composition factor. $\endgroup$– Geoff RobinsonSep 26, 2015 at 7:14
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$\begingroup$ Thank you Geoff. It helps to have a better understanding of the group-theoretic situation. $\endgroup$– David TowersSep 26, 2015 at 8:29
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$\begingroup$ Forster uses this construction to show that F*(F*(G)) < F*(G) where F*(G) is a generalised Fitting subgroup (not the quasinilpotent radical). I find his paper hard to follow as my German is very poor. A similar construction for Lie algebras, which looks possible using projective covers, could produce a Lie algebra with interesting properties. $\endgroup$– David TowersSep 26, 2015 at 10:14
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