A generalization of the Chermak--Delgado measure on subgroups and its associated lattice

Document Type : Ischia Group Theory 2022

Authors

1 School of Computer and Cyber Sciences, Augusta University, 100 Grace Hopper Ln, Augusta, GA 30901, USA

2 Department of Mathematics and Statistics, Binghamton University, P.O. Box 6000, Binghamton, NY 13902-6000, USA

3 Department of Mathematics, University of Louisiana at Lafayette, P.O. Box 43568, Lafayette LA 70504-3568 USA

10.22108/ijgt.2023.138469.1859

Abstract

We generalize the Chermak--Delgado measure of a subgroup of a finite group $G$, $\mu(H) = |H||C_{G}(H)|$, and its associated lattice of subgroups with maximal measure. We consider mappings $M$ of the lattice of all subgroups $\mathrm{Sub}(G)$ into itself and define a measure associated to $M$ by setting $\mu(H)=|H||M(H)|$. We investigate under what conditions on $M$ the subgroups with maximal measure form a sublattice of $\mathrm{Sub}(G)$. In particular, our focus is on the case where $M(H)$ is a centralizer-like subgroup.

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References

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International Journal of Group Theory
Volume 14, Issue 2 - Serial Number 2
Proceedings of the Ischia Group Theory 2022-Part 3
June 2025
Pages 75-91

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History

  • Receive Date: 20 July 2023
  • Revise Date: 23 November 2023
  • Accept Date: 01 December 2023
  • Published Online: 16 December 2023

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