On the Occurrence of the Complete Graph in the Hasse Graph of a Finite Group
Let G be any finite group and L(G) the lattice of all subgroups of G. If L(G) is strongly balanced (globally permutable) then we observe that the uniform dimension and the strong uniform dimension of L(G) are well defined, and we show how to calculate these dimensions.
We introduce and study the lattice of normal subgroups of a group G that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of G, see [Kaplan G., Levy D., Solitary subgroups, Comm. Algebra, 2009, 37(6), 1873–1883]. A precise description of this lattice is given for some particular classes of finite groups.
In this paper we study finite non abelian groups in which every proper normal subgroup and every proper epimorphic image is abelian. Also we study finite non nilpotent groups in which every normal subgroup and every proper epimorphic image is nilpotent and those finite soluble non nilpotent groups in which every proper normal subgroup is nilpotent.
Groups all whose nonidentity subgroups split over a normal inseparable nonidentity subgroup are studied.
In this paper we study the class of finite groups whose nilpotent residual is a Hall subgroup having all subgroups normal in .