%0 Journal Article
%T Groups in which every subgroup has finite index in its Frattini closure
%J Bulletin of the Iranian Mathematical Society
%I Iranian Mathematical Society (IMS)
%Z 1017-060X
%A De Giovanni, F.
%A Imperatore, D.
%D 2014
%\ 10/01/2014
%V 40
%N 5
%P 1213-1226
%! Groups in which every subgroup has finite index in its Frattini closure
%K Maximal subgroup
%K Frattini closure
%K $FM$-group
%R
%X In 1970, Menegazzo [Gruppi nei quali ogni sottogruppo e intersezione di sottogruppi massimali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 48 (1970), 559--562.] gave a complete description of the structure of soluble $IM$-groups, i.e., groups in which every subgroup can be obtained as intersection of maximal subgroups. A group $G$ is said to have the $FM$-property if every subgroup of $G$ has finite index in the intersection $\hat X$ of all maximal subgroups of $G$ containing $X$. The behaviour of (generalized) soluble $FM$-groups is studied in this paper. Among other results, it is proved that if~$G$ is a (generalized) soluble group for which there exists a positive integer $k$ such that $|\hat X:X|\leq k$ for each subgroup $X$, then $G$ is finite-by-$IM$-by-finite, i.e., $G$ contains a finite normal subgroup $N$ such that $G/N$ is a finite extension of an $IM$-group.
%U http://bims.iranjournals.ir/article_561_3a8e1901acbb88df60bd42b42c8fb15d.pdf