Abstract: | Let D be a subgroup of the group G. The lattice of intermediate subgroups is studied. The subgroup F (D F G) is said to be D-complete, if DF=u:u F>=F. Let F be the subset of all D-complete intermediate subgroups. The system {F, NG(F)} is a fan for D in G (RZhMat, 1980, 5A208) if and only if Dx> is a D-complete subgroup for any x G. The set {Fga} coincides with the collection of subgroups of the form DA (1 A C G) if and only if for any x G the subgroup D, Dx is D-complete. The last condition holds, for example, for a pronormal subgroup D.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 13–19, 1980. |