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.
- M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Algebra 92 (1985), no. 1, 44–80. MR 772471, DOI https://doi.org/10.1016/0021-8693%2885%2990145-0
- Robert Baddeley and Andrea Lucchini, On representing finite lattices as intervals in subgroup lattices of finite groups, J. Algebra 196 (1997), no. 1, 1–100. MR 1474164, DOI https://doi.org/10.1006/jabr.1997.7069
- Michael Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 1986. MR 895134
- Pinhas Grossman and Vaughan F. R. Jones, Intermediate subfactors with no extra structure, J. Amer. Math. Soc. 20 (2007), no. 1, 219–265. MR 2257402, DOI https://doi.org/10.1090/S0894-0347-06-00531-5
- George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403–420. MR 202822, DOI https://doi.org/10.1016/0021-8693%2866%2990030-5
- Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592
- F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, DOI https://doi.org/10.2307/1968693
- Péter Pál Pálfy and Pavel Pudlák, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis 11 (1980), no. 1, 22–27. MR 593011, DOI https://doi.org/10.1007/BF02483080
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
- Yasuo Watatani, Lattices of intermediate subfactors, J. Funct. Anal. 140 (1996), no. 2, 312–334. MR 1409040, DOI https://doi.org/10.1006/jfan.1996.0110
[AS]AS M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Alg. 92 (1985), 44–80.
[BL]BL R. Baddeley and A. Lucchini, On Representing finite lattices as intervals in subgroup lattices of finite groups, J. Alg. 196 (1997), 1–100.
[FGT]FGT M. Aschbacher, Finite Group Theory, Cambridge Univ. Press, Cambridge, 1986.
[GJ]GJ P. Grossman and V. Jones, Intermediate subfactors with no extra structure, J. Amer. Math. Soc. 20 (2007), 219–265.
[Gl]Gl G. Glauberman, Central elements of core-free groups, J. Alg. 4 (1966), 403–420.
[GLS3]GLS3 D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Number 3, Mathematical Surveys and Mongraphs, vol. 40, American Math. Soc., Providence, 1999.
[MN]MN F. Murry and J. von Neumann, On rings of operators, Annals of Math. 37 (1936), 116–229.
[PP]PP P. Palfy and P. Pudlak, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis 11 (1980), 22–27.
[T]T M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York, 1979.
[W]W Y. Watatani, Lattices of intermediate subfactors, J. Functional Analysis 140 (1996), 312–334.
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.