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.
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 CG(A) act transitively on the maximal A-invariant q subgroups of G for any prime q≠p.
References
- Bender, Helmut; Glauberman, George (1994), Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 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