Finite groups with few subgroups not in the Chermak-Delgado lattice

TL;DR

This paper determines finite groups with few subgroups outside the Chermak–Delgado lattice by classifying groups according to $v(G)$, the number of conjugacy classes of subgroups not in $\mathcal{CD}(G)$. It leverages Chermak–Delgado measure $m_G(H)$ and lattice properties, together with known results on $p$-groups (including generalized quaternion groups and $M_{p^n}$) and structural lemmas, to prove that $v(G)=1$ characterizes $G\cong Z_p$ or $Q_8$, $v(G)=2$ characterizes $G\cong Z_{q^2}$ or $M_{p^3}$ with $p>2$, and $v(G)=3$ (when $G$ is not nilpotent) characterizes nonabelian groups of order $pq$ with $p\ne q$. The results refine understanding of how the Chermak–Delgado lattice constrains subgroup structure in finite groups, with implications for identifying groups with minimal CD-exceptions.

Abstract

For a finite group G, we denote by v(G) the number of conjugacy classes of subgroups of G not in CD(G). In this paper, we determine the finite groups G such that v(G)=1,2,3.

Theorems & Definitions (16)

• Theorem 1.1
• proof
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• proof
• Corollary 2.3
• Proposition 2.4
• proof
• Theorem 2.5