Some results about a theorem of Shidov

Abstract

The authors characterize the finite groups in which $\,\mathcal{H} (G)$ , the intersection of the maximal non-nilpotent subgroups of G, is nilpotent, but different from Φ(G). Further, if $\,\mathcal{F} \,$ is a saturated formation and if $\,\mathcal{F}(G)\,$ is the intersection of all maximal subgroups of G not belonging to $\,\mathcal{F}$ , a necessary and sufficient condition is given for $\,\mathcal{F}(G)\,$ to be nilpotent different from Φ(G). Keywords: Frattini subgroup, Maximal subgroups, Saturated formation Mathematics Subject Classification (2000): 20B05, 20D10, 20D25,20E28