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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: [email protected])
ELIZABETH WILCOX*
Affiliation:
Department of Mathematics, Colgate University, 13 Oak Drive, Hamilton, NY 13346, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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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

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