Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T09:07:31.106Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME GROUPS WITH COMPUTABLE CHERMAK–DELGADO LATTICES

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  27 March 2012

BEN BREWSTER  and
ELIZABETH WILCOX
BEN BREWSTER
Affiliation:
Department of Mathematical Sciences, Binghamton University, Vestal, NY 13902-6000, USA (email: ben@math.binghamton.edu)
ELIZABETH WILCOX*
Affiliation:
Department of Mathematics, Colgate University, 13 Oak Drive, Hamilton, NY 13346, USA (email: ewilcox@colgate.edu)
*
✉For correspondence; e-mail: ewilcox@colgate.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a finite group and let H≤G. We refer to |H||CG(H)| as the Chermak–Delgado measure ofH with respect to G. Originally described by Chermak and Delgado, the collection of all subgroups of G with maximal Chermak–Delgado measure, denoted 𝒞𝒟(G), is a sublattice of the lattice of all subgroups of G. In this paper we note that if H∈𝒞𝒟(G) then H is subnormal in G and prove that if K is a second finite group then 𝒞𝒟(G×K)=𝒞𝒟(G)×𝒞𝒟(K) . We additionally describe the 𝒞𝒟(G≀Cp) where G has a nontrivial centre and p is an odd prime and determine conditions for a wreath product to be a member of its own Chermak–Delgado lattice. We also examine the behaviour of centrally large subgroups, a subset of the Chermak–Delgado lattice.

Keywords

subgroup latticessubnormal subgroups

MSC classification

Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 29 - 40
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Chermak, A. and Delgado, A., ‘A measuring argument for finite groups’, Proc. Amer. Math. Soc. 107(4) (1989), 907–914.CrossRefGoogle Scholar
[2]Dummit, D. S. and Foote, R. M., Abstract Algebra, 3rd edn (Wiley, Hoboken, NJ, 2004).Google Scholar
[3] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008).Google Scholar
[4]Glauberman, G., ‘Centrally large subgroups of finite p-groups’, J. Algebra 300 (2006), 480–508.CrossRefGoogle Scholar
[5]Isaacs, I. M., Finite Group Theory (American Mathematical Society, Providence, RI, 2008).Google Scholar
[6]Thwaites, G. N., ‘The abelian p-subgroups of GL n(p) of maximal rank’, Bull. Lond. Math. Soc. 4 (1972), 313–320.CrossRefGoogle Scholar
[7]Wilcox, E., Complete Finite Frobenius Groups and Wreath Products. PhD Thesis, Binghamton University, 2010.Google Scholar