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A pair of characteristic subgroups for pushing-up. II

Part of: Structure and classification of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  24 October 2012

George Glauberman
George Glauberman*
Affiliation:
Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514, USA (gg@math.uchicago.edu)
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Abstract

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Many problems about local analysis in a finite group G reduce to a special case in which G has a large normal p-subgroup satisfying several restrictions. In 1983, R. Niles and G. Glauberman showed that every finite p-group S of nilpotence class at least 4 must have two characteristic subgroups S1 and S2 such that, whenever S is a Sylow p-subgroup of a group G as above, S1 or S2 is normal in G. In this paper, we prove a similar theorem with a more explicit choice of S1 and S2.

Keywords

Sylow p-subgroupscharacteristic subgroup

MSC classification

Secondary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure 20E99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 71 - 133
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Aschbacher, M., Generation of fusion systems of characteristic-2 type, Invent. Math. 180 (2010), 225–299.CrossRefGoogle Scholar
2.Baumann, B., Über endliche Gruppen mit einer zu L2(2n) isomorphen Faktorgruppe, Proc. Am. Math. Soc. 74 (1979), 215–222.Google Scholar
3.Campbell, N., Pushing-up in finite groups, PhD Thesis, California Institute of Technology, Pasadena, CA (1979).Google Scholar
4.Carter, R., Simple groups of Lie type (Wiley, 1972).Google Scholar
5.Chermak, A. and Delgado, A., A measuring argument for finite groups, Proc. Am. Math. Soc. 107 (1989), 907–914.CrossRefGoogle Scholar
6.Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Wiley, 1962).Google Scholar
7.Feit, W., Characters of finite groups (W. A. Benjamin, New York, 1967).Google Scholar
8.Glauberman, G., A characteristic subgroup of a p-stable group, Can. J. Math. 20 (1968), 1101–1135.CrossRefGoogle Scholar
9.Glauberman, G., Global and local properties of finite groups, in Finite simple groups (ed. Powell, M. B. and Higman, G.), pp. 1–64 (Academic Press, New York, 1971).Google Scholar
10.Glauberman, G., Isomorphic subgroups of finite p-subgroups, I, Can. J. Math. 23 (1971), 983–1022.CrossRefGoogle Scholar
11.Glauberman, G., Centrally large subgroups of finite p-groups, J. Alg. 300 (2006), 480–508.CrossRefGoogle Scholar
12.Glauberman, G. and Niles, R., A pair of characteristic subgroups for pushing-up in finite groups, Proc. Lond. Math. Soc. 46 (1983), 411–453.CrossRefGoogle Scholar
13.Gorenstein, D., Finite groups (Harper & Row, New York, 1968).Google Scholar
14.Gorenstein, D., Finite simple groups: an introduction to their classification (Plenum, New York, 1982).CrossRefGoogle Scholar
15.Griess, R. L. Jr, Schur multipliers of finite simple groups of Lie type, Trans. Am. Math. Soc. 183 (1973), 355–421.CrossRefGoogle Scholar
16.Hall, M., The theory of groups (Macmillan, New York, 1959).Google Scholar
17.Huppert, B., Endliche Gruppen I (Springer, 1967).CrossRefGoogle Scholar
18.Khukhro, E. I., p-automorphisms of finite p-groups, London Mathematical Society Lecture Note Series, Volume 246 (Cambridge University Press, 1998).CrossRefGoogle Scholar
19.Niles, R., Pushing-up in finite groups, J. Alg. 57 (1979), 26–63.CrossRefGoogle Scholar
20.Richen, F., Modular representations of split BN pairs, Trans. Am. Math. Soc. 140 (1969), 435–460.Google Scholar
21.Rose, J. S., A course on group theory (Dover, New York, 1994).Google Scholar
22.Steinberg, R., Lectures on Chevalley groups, Yale University (1968) (available at www.math.ucla.edu/~rst).Google Scholar