Le G be a finite group. Following Schmid [18], whenever θ is a non-empty set of subgroups of G, we write θ~⊥ for the set of subgroups H of G such that HT = TH for all T ∈θ, that is, such that H permutes with all subgroups belonging to θ. One possible choice of θ is Syl(G), the class of all Sylow subgroups of G. Subgroups of G belonging to Syl(G)~⊥ have been studied in [2], [7], [11], [18]. By results of Kegel [11] and Schmid [18], Syl(G)~⊥ is a sublattice of the lattice S_n(G) of subnormal subgroups of G. In the sequel all groups are understood to be finite.
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