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

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics and Statistics, Binghamton University, Binghamton, USA

2 Faculty of Mathematics, “Al. I. Cuza” University, Ia¸si, Romania

Abstract

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

Keywords

Main Subjects

References

[1] L. An, J. P. Brennan, H. Qu and E. Wilcox, Chermak–Chermak-Delgado lattice extension theorems, Comm. Algebra, 43 no. 5 (2015) 2201–2213.
[2] B. Brewster and E. Wilcox, Some groups with computable Chermak-Delgado lattices, Bull. Aust. Math. Soc., 86 no. 1 (2012) 29–40.
[3] B. Brewster, P. Hauck and E. Wilcox, Groups whose Chermak-Delgado lattice is a chain, J. Group Theory, 17 no. 2 (2014) 253–265.
[4] A. Chermak and A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc., 107 no. 4 (1989) 907–914.
[5] A. Galoppo, Groups with dense normal-by-finite subgroups, Ricerche Mat., 46 no. 1 (1997) 45–48.
[6] A. Galoppo, Groups with dense nearly normal subgroups, Note Mat., 20 (2000/2001) 15–19.
[7] F. de Giovanni and A. Russo, Groups with dense subnormal subgroups, Rend. Sem. Mat. Univ. Padova, 101 (1999) 19–27.
[8] I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI, 2008.
[9] A. Mann, Groups with dense normal subgroups, Israel J. Math., 6 (1968) 13–25.
[10] R. McCulloch, Chermak–Delgado simple groups, Comm. Algebra, 45 (2017) 983–991.
[11] R. McCulloch, Finite groups with a trivial Chermak-Delgado subgroup, J. Group Theory, 21 no. 3 (2018) 449–461.
[12] R. McCulloch and M. T˘arn˘auceanu, Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero, Comm. Algebra, 46 no. 7 (2018) 3092–3096.
[13] R. McCulloch and M. T˘arn˘auceanu, On the Chermak-Delgado lattice of a finite group, Comm. Algebra, 48 no. 1 (2020) 37–44.
[14] A. Morresi Zuccari, V. Russo and C. M. Scoppola, The Chermak-Delgado measure in finite p-groups, J. Algebra, 502 (2018) 262–276.
[15] R. Schmidt, Subgroup lattices of groups, De Gruyter Expositions in Mathematics, 14, Walter de Gruyter & Co., Berlin, 1994.
[16] M. T˘arn˘auceanu, A note on the Chermak–Delgado lattice of a finite group, Comm. Algebra, 46 (2018) 201–204.
[17] M. T˘arn˘auceanu, Finite groups with a certain number of values of the Chermak-Delgado measure, J. Algebra Appl., 19 no. 5 (2020) 7 pp.
[18] L. A. Sordo Vieira, On P-Adic Fields and P-Groups, Thesis (Ph.D.)–University of Kentucky, ProQuest LLC, Ann Arbor, MI, 2017.
[19] G. Vincenzi, Groups with dense pronormal subgroups, Ricerche Mat., 40 no. 1 (1991) 75–79.
[20] E. Wilcox, Exploring the Chermak-Delgado lattice, Math. Mag., 89 no. 1 (2016) 38–44.
International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 12 April 2025

Files

History

  • Receive Date: 15 January 2025
  • Revise Date: 18 March 2025
  • Accept Date: 23 March 2025
  • Published Online: 12 April 2025

Share

How to cite

Statistics