Abstract
In this note we prove that the Chermak–Delgado lattice of a ZM-group is a chain of length 0, more precisely \({\mathcal {CD}}(\mathrm{ZM}(m,n,r))=\{H_{(1,d,0)}\}\), where d is the multiplicative order of r modulo m. A similar conclusion is obtained for all dihedral groups \(D_{2m}\) with \(m\ne 4\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An, L., Brennan, J.P., Qu, H., Wilcox, E.: Chermak–Delgado lattice extension theorems. Comm. Algebra 43, 2201–2213 (2015)
Brandl, R., Cutolo, G., Rinauro, S.: Posets of subgroups of groups and distributivity. Boll. U.M.I. 9–A, 217–223 (1995)
Brewster, B., Wilcox, E.: Some groups with computable Chermak–Delgado lattices. Bull. Aust. Math. Soc. 86, 29–40 (2012)
Brewster, B., Hauck, P., Wilcox, E.: Groups whose Chermak–Delgado lattice is a chain. J. Gr. Theory 17, 253–279 (2014)
Calhoun, W.C.: Counting subgroups of some finite groups. Am. Math. Mon. 94, 54–59 (1987)
Chermak, A., Delgado, A.: A measuring argument for finite groups. Proc. AMS 107, 907–914 (1989)
Huppert, B.: Endliche Gruppen, I. Springer, Berlin (1967)
Isaacs, I.M.: Finite Group Theory. American Mathematical Society, Providence (2008)
Tărnăuceanu, M.: The normal subgroup structure of ZM-groups. Ann. Mat. Pura Appl. 193, 1085–1088 (2014)
Wilcox, E.: Exploring the Chermak–Delgado lattice. Math. Mag. 89, 38–44 (2016)
Acknowledgements
The author is grateful to the reviewer for its remarks which improve the previous version of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tărnăuceanu, M. The Chermak–Delgado Lattice of ZM-Groups. Results Math 72, 1849–1855 (2017). https://doi.org/10.1007/s00025-017-0735-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-017-0735-z