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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

On intervals in subgroup lattices of finite groups


Author: Michael Aschbacher
Journal: J. Amer. Math. Soc. 21 (2008), 809-830
MSC (2000): Primary 20D30; Secondary 06B05, 46L37
Published electronically: March 17, 2008
MathSciNet review: 2393428
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Abstract: We investigate the question of which finite lattices $ L$ are isomorphic to the lattice $ [H,G]$ of all overgroups of a subgroup $ H$ in a finite group $ G$. We show that the structure of $ G$ is highly restricted if $ [H,G]$ is disconnected. We define the notion of a ``signalizer lattice" in $ H$ and show for suitable disconnected lattices $ L$, if $ [H,G]$ is minimal subject to being isomorphic to $ L$ or its dual, then either $ G$ is almost simple or $ H$ admits a signalizer lattice isomorphic to $ L$ or its dual. We use this theory to answer a question in functional analysis raised by Watatani.


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Additional Information

Michael Aschbacher
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125

DOI: http://dx.doi.org/10.1090/S0894-0347-08-00602-4
PII: S 0894-0347(08)00602-4
Received by editor(s): June 28, 2006
Published electronically: March 17, 2008
Additional Notes: This work was partially supported by NSF-0504852
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.