$M$-subgroups and normal-subgroup lattices of finite groups

Abstract

Let $G$ be a group. A normal subgroup $E$ is an $M$-subgroup if $E\cap N_1\cap N_2\neq1$ whenever $N_1\cap N_2$, $E\cap N_1$, and $E\cap N_2$ are non-trivial for normal subgroups $N_1,N_2$. We characterise these subgroups as the essential or relatively uniform normal subgroups. For finite $G$, an $M$-subgroup is either trivial, contains exactly one minimal normal subgroup, or contains the socle. Consequently, every normal subgroup is an $M$-subgroup precisely when the socle has $G$-chief length at most two. We obtain an explicit $M$-closure operation and describe the minimal normal subgroups of finite direct products. For crown-based powers, we use the known comparability of normal subgroups with the socle to describe the normal lattice in terms of Boolean or subspace lattices and the quotient above the socle. We then determine when the $M$-lattice retains the normal lattice, classify normal quotients and their automorphism orbits, and characterise quotient maps compatible with $M$-closure. Finally, we compute the maximal length of chains of $M$-subgroups in terms of the chief length and socle length of $G$.