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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Detecting the index of a subgroup in the subgroup lattice


Authors: M. De Falco, F. de Giovanni, C. Musella and R. Schmidt
Journal: Proc. Amer. Math. Soc. 133 (2005), 979-985
MSC (2000): Primary 20E15
Published electronically: September 16, 2004
MathSciNet review: 2117197
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Abstract: A theorem by Zacher and Rips states that the finiteness of the index of a subgroup can be described in terms of purely lattice-theoretic concepts. On the other hand, it is clear that if $G$ is a group and $H$ is a subgroup of finite index of $G$, the index $\vert G:H\vert$ cannot be recognized in the lattice ${\mathfrak{L}}(G)$ of all subgroups of $G$, as for instance all groups of prime order have isomorphic subgroup lattices. The aim of this paper is to give a lattice-theoretic characterization of the number of prime factors (with multiplicity) of $\vert G:H\vert$.


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Additional Information

M. De Falco
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, Via Cintia, I - 80126 Napoli, Italy
Email: mdefalco@unina.it

F. de Giovanni
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, Via Cintia, I - 80126 Napoli, Italy
Email: degiovan@unina.it

C. Musella
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, Via Cintia, I - 80126 Napoli, Italy
Email: cmusella@unina.it

R. Schmidt
Affiliation: Mathematisches Seminar, Universität Kiel, Ludwig-Meyn Straße 4, D - 24098 Kiel, Germany
Email: schmidt@math.uni-kiel.de

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07638-5
PII: S 0002-9939(04)07638-5
Received by editor(s): October 8, 2003
Received by editor(s) in revised form: December 1, 2003
Published electronically: September 16, 2004
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.