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SOME GROUPS WITH COMPUTABLE CHERMAK–DELGADO LATTICES

  • BEN BREWSTER (a1) and ELIZABETH WILCOX (a2)

Abstract

Let G be a finite group and let HG. 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(GCp) 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.

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Copyright

Corresponding author

For correspondence; e-mail: ewilcox@colgate.edu

References

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[1]Chermak, A. and Delgado, A., ‘A measuring argument for finite groups’, Proc. Amer. Math. Soc. 107(4) (1989), 907914.
[2]Dummit, D. S. and Foote, R. M., Abstract Algebra, 3rd edn (Wiley, Hoboken, NJ, 2004).
[3] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008).
[4]Glauberman, G., ‘Centrally large subgroups of finite p-groups’, J. Algebra 300 (2006), 480508.
[5]Isaacs, I. M., Finite Group Theory (American Mathematical Society, Providence, RI, 2008).
[6]Thwaites, G. N., ‘The abelian p-subgroups of GL n(p) of maximal rank’, Bull. Lond. Math. Soc. 4 (1972), 313320.
[7]Wilcox, E., Complete Finite Frobenius Groups and Wreath Products. PhD Thesis, Binghamton University, 2010.
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SOME GROUPS WITH COMPUTABLE CHERMAK–DELGADO LATTICES

  • BEN BREWSTER (a1) and ELIZABETH WILCOX (a2)

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