Finite groups with dense ${\cal CD}$-subgroups

TL;DR

We study finite groups with dense $\mathcal{CD}$-subgroups, meaning every nonempty open interval of the subgroup lattice $L(G)$ contains a subgroup in $\mathcal{CD}(G)$. For finite $p$-groups of order $p^n$ with this density, we obtain $|Z(G)|=p$, $m^*(G)=p^{n+1}$, every subgroup of order $p^2$ contains $Z(G)$, and $\mathrm{Im}(m_G)=\{p^n,p^{n+1}\}$, with normal subgroups of order $p^2$ lying in $\mathcal{CD}(G)$. If $G$ is not a $p$-group, the density property holds precisely for nonabelian groups of order $pq$. Extraspecial $p$-groups exhibit a nuanced dependence on the prime and the exponent: for odd $p$, all extraspecial groups of order $p^3$ have dense $\mathcal{CD}$-subgroups, while higher exponents typically do not; for $p=2$ there is a similar dichotomy among order $32$ extraspecial groups. These results yield sharp structural constraints on the center and the Chermak–Delgado lattice across both $p$-group and non-$p$-group regimes, clarifying when density phenomena occur in finite groups.

Abstract

A group $G$ is said to have dense ${\cal CD}$-subgroups if each non-empty open interval of the subgroup lattice $L(G)$ contains a subgroup in the Chermak--Delgado lattice ${\cal CD}(G)$. In this note, we study finite groups satisfying this property.

Theorems & Definitions (2)

• Theorem 1.1
• Theorem 1.2