Finite groups with many normalizers
Marius Tărnăuceanu
TL;DR
The paper studies finite groups through the lens of normalizers in the subgroup lattice, proving that dense normalizers occur precisely for groups that are cyclic of prime order or non-abelian of order $pq$. It then classifies groups by the deficit $|L(G)|-|N_G|$, giving complete lists for deficits $k=1,2,3$, and, for non-$ZM$ groups, deficit $k=4$, with explicit isomorphism types including $\mathbb{Z}_p$, $\mathbb{Z}_{p^2}$, $\mathbb{Z}_p\rtimes\mathbb{Z}_q$ ($q\mid p-1$), $\mathbb{Z}_{p^3}$, $\mathbb{Z}_{pq}$, $\mathbb{Z}_{p^2}\rtimes\mathbb{Z}_q$, $\mathbb{Z}_{p^4}$, $\mathbb{Z}_2\times\mathbb{Z}_2$, and $A_4$, with $A_4$ uniquely characterized by the deficit among groups with a non-cyclic Sylow subgroup. ZM-groups with deficit 4 are classified via $\tau(m)+\tau(n)\le6$, yielding examples like $D_{30}$ and $D_{54}$. These results deepen understanding of how normalizers populate subgroup lattices and provide explicit finite-group classifications useful for algebraic and computational investigations.
Abstract
A group $G$ is said to have dense normalizers if each non-empty open interval in its subgroup lattice $L(G)$ contains the normalizer of a certain subgroup of $G$. In this note, we find all finite groups satisfying this property. We also classify the finite groups in which $k$ subgroups are not normalizers, for $k=1,2,3,4$.