Thompson transitivity theorem

In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It originated in the proof of the odd order theorem by Feit and Thompson (1963), where it was used to prove the Thompson uniqueness theorem.

Statement

Suppose that G is a finite group and p a prime such that all p-local subgroups are p-constrained. If A is a self-centralizing normal abelian subgroup of a p-Sylow subgroup such that A has rank at least 3, then the centralizer C_G(A) act transitively on the maximal A-invariant q subgroups of G for any prime q ≠ p.

gollark: Someone made an AR game.

gollark: And handle mouse clicks.

gollark: 🌵 🐪 Anyway, I'm not sure how you plan to route events to the right terminal.

gollark: You cannot escape the terminals.

gollark: You need to do .canvas3d() to get the 3D canvas to put it on, and I think something with an origin?

References

Bender, Helmut; Glauberman, George (1994), Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series, 188, Cambridge University Press, ISBN 978-0-521-45716-3, MR 1311244
Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order", Pacific Journal of Mathematics, 13: 775–1029, ISSN 0030-8730, MR 0166261
Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR 0569209

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.