## On intervals in subgroup lattices of finite groups

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- by Michael Aschbacher PDF
- J. Amer. Math. Soc.
**21**(2008), 809-830 Request permission

## 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.## References

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## Additional Information

**Michael Aschbacher**- Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 27630
- Received by editor(s): June 28, 2006
- Published electronically: March 17, 2008
- Additional Notes: This work was partially supported by NSF-0504852
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**21**(2008), 809-830 - MSC (2000): Primary 20D30; Secondary 06B05, 46L37
- DOI: https://doi.org/10.1090/S0894-0347-08-00602-4
- MathSciNet review: 2393428