A characteristic subgroup for fusion systems

Details A characteristic subgroup for fusion systems A characteristic subgroup for fusion systems

2008-11-22

arxiv.org/abs/0811.3666v1

Journal of Algebra 322 (2009) 1705-1718

Mathematics

Representation Theory

LaTeX file, 19 pages

Scientific paper

10.1016/j.jalgebra.2009.05.026

As a counterpart for the prime 2 to Glauberman's $ZJ$-theorem, Stellmacher proves that any nontrivial 2-group $S$ has a nontrivial characteristic subgroup $W(S)$ with the following property. For any finite $\Sigma_4$-free group $G$, with $S$ a Sylow 2-subgroup of $G$ and with $O_2(G)$ self-centralizing, the subgroup $W(S)$ is normal in $G$. We generalize Stellmacher's result to fusion systems. A similar construction of $W(S)$ can be done for odd primes and gives rise to a Glauberman functor.

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