On groups with many maximal subgroups (Q2644764)

A group G is an IM-group if every subgroup of G is an intersection of maximal subgroups of G. The class of IM-groups has been studied by \textit{F. Menegazzo} [Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat.., VIII. Ser. 48, 559-562 (1970; Zbl 0216.088)]. Generalizing this concept, the author defines an IIM-group as a group in which every infinite subgroup is an intersection of maximal subgroups. He proves that an infinite group is a soluble IIM-group if and only if it is either a soluble IM-group or an extension of a Prüfer group by a finite IM- group. Moreover, if G is an infinite IIM-group having an ascending or a descending series with finite or locally nilpotent factors, then G is soluble. A group in which every subnormal subgroup (respectively: every infinite subnormal subgroup) is an intersection of maximal subgroups is called in IMsn-group (respectively: an IIMsn-group). It is proved that for soluble groups the properties IMsn and IIMsn coincide with the properties IM and IIM, respectively. Finally, a group is said to be just-non-IM if it is not an IM-group but all its proper quotients are IM-groups. The author completely classifies soluble just-non-IM groups.