Let R be a p -group, and Σ a set of groups such that for all L in Σ , R is a subgroup of L, R=O_{ p} (N_{L}(R)), and such that N_{L}(R) contains a point stabilizer of L. ( If T is a Sylow p-subgroup of L, then O^{ p'} (C_{ L}(Ω_{1}Z(T))) is called a point stabilizer of L). In this paper we classify the pairs (R,Σ) such that no non-trivial subgroup of R is normal in all the L's from Σ.

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