Finite groups with the most Chermak-Delgado measures of subgroups
Guojie Liu, Haipeng Qu, Lijian An
TL;DR
The paper investigates the Chermak-Delgado measures $m_G(H)=|H|\cdot|C_G(H)|$ on finite groups, focusing on the number of distinct measures $|\mathrm{Im}(m_G)|$ and the maximal possible value $|\mathrm{Im}_{\max}(n)|$ for groups of order $n$. It establishes precise $|\mathrm{Im}_{\max}(n)|$ results for $n=p^k$ across three regimes and extends to nilpotent and square-free orders using direct-product decompositions, characterizing when equality with the maximal bound occurs (abelian groups, maximal-class groups with a uniform element, etc.). The work further derives general upper bounds for $|\mathrm{Im}(m_G)|$ tied to the group’s structure, analyzes the effect of coprime-direct products and semidirect products, and provides detailed computations for dihedral groups and a comprehensive $n=60$ example to illustrate the landscape and pose open questions. Collectively, the results connect subgroup lattices, central series, and group-class properties to quantify the diversity of Chermak-Delgado measures. The study advances understanding of how group structure governs the Chermak-Delgado lattice and its image, with implications for classifying groups by their measure distributions.
Abstract
Let $G$ be a finite group and $H\leq G$. The Chermak-Delgado measure of $H$ is defined as the number $|H|\cdot|C_{G}(H)|$. In this paper, we identify finite groups that exhibit the maximum number of Chermak-Delgado measures under some specific conditions.