De Giovanni, F., Imperatore, D. (2014). Groups in which every subgroup has finite index in its Frattini closure. Bulletin of the Iranian Mathematical Society, 40(5), 1213-1226.
F. De Giovanni; D. Imperatore. "Groups in which every subgroup has finite index in its Frattini closure". Bulletin of the Iranian Mathematical Society, 40, 5, 2014, 1213-1226.
De Giovanni, F., Imperatore, D. (2014). 'Groups in which every subgroup has finite index in its Frattini closure', Bulletin of the Iranian Mathematical Society, 40(5), pp. 1213-1226.
De Giovanni, F., Imperatore, D. Groups in which every subgroup has finite index in its Frattini closure. Bulletin of the Iranian Mathematical Society, 2014; 40(5): 1213-1226.
Groups in which every subgroup has finite index in its Frattini closure
2Università di Napoli "Federico II" Dipartimento di Matematica e Applicazioni
Receive Date: 12 July 2013,
Revise Date: 08 September 2013,
Accept Date: 09 September 2013
Abstract
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.
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